The reason why Entropy has so many descriptions is not because it was designed to. Nobody started out with all of those things called Entropy.
Entropy started off with one thing. And then a bunch of other stuff was found to be related, both mathematically and physically, to that one thing.
Way back, it was observed that all useful energy ends up turning into useless and diffuse heat. This process was called Entropy. They knew it happened. They didn't know why.
So that is the start. Entropy, when originally named, was just a description of something that happens whenever you poke at the universe and look at what happens. That Energy becomes useless waste heat.
Every other definition of Entropy is because either it was a way to describe why or how that happens, or because the mathematics lined up with other Entropy mathematics. And surprisingly often that lining up of mathematics actually ends up having a physical meaning.
This is known as the unreasonable effectiveness of mathematics in the natural sciences; mathematical patterns keep on explaining things about the universe, and naively this is very surprising and unreasonable.
So, going back to Entropy. We start with observational science. That waste heat thing:
It was observed that heat flows from hot things to cold things. This was mathematically modeled with values like temperature and heat flow. This is a law of Entropy, that heat energy flows from hot things to cold things and not other way around. From it you can generate an insane amount of descriptive power about the universe.
Along comes Boltzman, who takes that "heat flow" and describes it more abstractly. He describes "macrostates"; something we can describe at our chosen scale. You divide your description of what could happen into a bunch of macrostates (as many or as few as you want).
In each macrostate there are many, many indistinguishable (at our scale) "microstates" that produce the "same" macrostate.
A microstate is a microstate because, while it is different than other microstates, it is in a way that we don't care about when we originally described our macrostates.
For example, "there is a table in the room" is a macrostate. The scratches on the table, the exact location of the termite in it, the current velocity of the atom of carbon in the geometric center of the table -- all of those are not described by my macrostate.
So, we call all of the actual, physical states we clumped into that macrostate to be microstates.
If you count those microstates in each macrostate, you find that a closed system almost always moves into the more common "macrostates" from the rarer ones. And this is sufficient to describe the transfer of energy from hot objects to cold ones; the number of microstates in two tepid objects is insanely higher than the number of microstates that describe one hot and one cold object.
This is strange. But, as it turns out, that when we actually go and count how many states a given macrostate has, instead of getting "table in room has 10 billion states" and "rubble of table has 15 billion states" -- ie, the two numbers are relatively similar, we get something crazy like "rubble case has $10^{1000000}$ times as many microstatesthan table does". (the exact number is not accurate, the point is that it is a ridiculously huge factor, not a small one)
This is so true that we end up measuring the number of microstates by taking the logarithm. So we get the table has X Entropy, and the rubble has X+1000000 Entropy. Only a million more units of entropy; but because this is on an exponential scale, that is actually $10^{1000000}$ times more states.
This statistical description of Entropy matches the earlier one -- it explains why heat energy flows from hot to cold objects, and why useful energy ends up being emitted as "useless" diffuse waste heat.
Weird. But not weird enough. Now things get strange.
Way off in mathematics, someone was working on a subject called Information Theory. This is useful to do things like figure out how to pass more information along a wire or radio signal; how much can you send? Can you improve on this protocol with one that sends more? How do you fix errors caused by random noise? Given an English sentence or piece of music or a picture, how much can you compress it and still get the original back afterwards?
Shannon generated a measure of information in a system. And, somewhat amazingly, it ends up working like physical Entropy does; the same mathematical equations govern both of these. And, with work, you can connect Shannon information entropy to Boltzman statistical entropy in physical ways.
From there you get further abstractions and remixing. Things that "behave like" Entropy in a new domain are called Entropy. And often when connected back to macroscopic physics and the transfer of heat it is the same phenomena and deduces "energy tends to become useless, diffuse heat".
Now, part of your confusion is that you are looking back to the big bang, and saying "but that was a state of really low Entropy!".
And yes, it was. We are going to have a far lower entropy than the universe when Big Bang happened has.
Why did the Big Bang happen? That isn't explained by Entropy. Entropy tells us why the Big Bang leads to us, and why the Big Bang must be a point of extremely low Entropy. Not every piece of reality is explained by every piece of scientific theory. In order to investigate the Big Bang's "origins" you'll have to use more than the laws of Entropy.
Entropy applies to a closed system; parts of that system can have reduced Entropy, but only at the cost of increasing the Entropy elsewhere in the system more. Snowflakes or Humans are not contradictions of the laws of Entropy, because in both cases they formed as part of a larger system.
Also, your information description is backwards. Entropy is a measure of how much information it would require to fully describe a system, and it never decreases. This means that the Big Bang, as a low entropy state, is the simplest phase of the universe to fully describe.
Now, this "full description" tends to be extremely boring. You are doing something like describing the location and movement of every single particle, individually (I am ignoring QM here; it has its own definition of entropy that is consistent, but not something I'm going into here). When you have a room full of gas bouncing around at random, that is harder to describe than the same number of particles all arranged in a regular grid.
Suppose we take the room full of gas, freeze it, and carve a fancy sculpture out of the resulting crystal. To us, the shape that crystal takes is more interesting than the boring "room full of gas". But fully describing that crystal statue turns out to be insanely easier; the particles are more constrained in position and velocity, they form a regular grid instead of a chaotic gas. The exact shape of the crystal doesn't require all that much information, but constrains the number of states the atoms can be in a huge amount. It is a very low entropy state, when you fully describe everything.
We are just bored by the room of gas, but interested in the crystal statue, so we talk more about the statue than the room of gas.
Humans often like low-entropy things. Our brains are pattern-matchers, and low-entropy things have lots of patterns. High entropy things tend to be "boring" smears, as constraining things to a pattern is a massive reduction in the positions the atomic scale particles can be in.
Let's get concrete.
So why can't we unbreak an egg?
The "macro/macrostate" trick is a bit fun. Counter intuitively to us, the high entropy macrostates have insanely large numbers of microstates compared to the low entropy states. When we convert entropy to a number, we take the logarithm of the number of microstates. So every 'unit' of entropy is an exponential increase in the number of microstates.
A 'high' entropy situation could have an entropy thousands or millions of units greater; now take e
and raise it to a power of a million. That is how many more times greater the number of microstates the high entropy macrostate has.
If going from one state to another is anywhere close to uniform, going from a macrostate with X times $10^{1000000}$ more states back to a state with X states is going to be, well, not very likely. And that is what happens when you want to unbreak an egg. There are a simply ridiculous number of "broken egg" states, and very very few "unbroken egg" states. The dropping of the egg on the floor disrupts the somewhat stable "unbroken egg" macrostate, and moves it into a random state in the (broken egg + unbroken egg) combined state.
Going from the (unbroken egg + broken egg) combined state back to an unbroken egg requires that we reach one of those X states among the X times $10^{1000000}$ combined broken and unbroken states.
So now you feed the broken egg to a chicken (well, many broken eggs). And out comes a single unbroken egg. How?
The chicken takes the still-low-entropy molecules in the broken egg, and uses their "ordered" energy to order other molecules inside itself. This engine emits heat -- high entropy energy -- and concentrates some low entropy materials inside the chicken. Those low entropy materials are in turn converted into high entropy waste, and used to do build other low entropy materials (new cells, create membranes that concentrate calcium, make blood carrying sugars and oxygen go to a cell that will grow into an egg, transcribe DNA into RNA and RNA into proteins, etc).
After consuming a pile of low-entropy matter and converting it into higher-entropy heat and waste matter (poop!), it takes some matter and arranges it into an egg.
This process is not 100% efficient. That chicken produced more entropy in waste products than the difference between the raw materials and the finished egg has. A closed system with a chicken, which lays an egg, you then feed the egg back to the chicken, cannot produce new eggs without the chicken's biological engine damaging itself.
Typically the input to this process is via feeding the chicken plant materials, which in turn turned CO2 in the air into low entropy plant matter by absorbing low-entropy light from the sun.
The sun in turn produces light by taking low entropy hydrogen and fusing it into higher entropy helium. The pressure to do that was fueled by gravitational collapse, where a non-uniformity in interstellar gas caused some to clump, radiate heat as it fell in (that heat being high-entropy energy), pull in more gas, and grow until the center was hot and pressurized enough to start fusion.
The hydrogen fuel for the sun was left over from the low entropy big bang. Early in the big bang it was too hot for neutrons and protons to stay stuck together. As it cooled, they started fusing, but the rate of cooling was so fast that not all of the Hydrogen became Helium, and there wasn't enough time at the required pressure and temperature to fuse everything into Iron (the highest entropy atomic nucleus arrangement of neutrons and protons).
Imagine the world as an extremely steep slope that is also extremely, extremely long.
Bouncing down the slope are boulders. These boulders bounce off the ground, losing energy. As they do so they lose forward momentum.
But they are on a slope, so they also fall. This keeps them going.
Trying to get a boulder to roll uphill in the middle of this avalanche is insanely hard. Getting it to roll downhill is very easy.
Now, you could even use the boulder's to build a pattern, but that pattern has to roll down the hill as well; it can't stay stationary. The slope is too steep.
The universe, as best we can tell, is an extremely steep entropy slope from the big bang. We harvest leftover low entropy matter -- mostly hydrogen -- from the big bang, convert it to low entropy light, convert that to carbon plants, bury all of it and have it decay into hydrocarbon, burn those hydrocarbons to run our coal plants, make that vibrate electrons to produce electrical power, use that to convert aluminum ore into pure metal and run machines that stamp out cans of it, then open the can and drink some clean water from it.
Each of those harvests is like using the energy from one of those falling boulders (as we ourselves are also falling) in order to get things done.