What is temperature and what do degrees mean? This stems from a question I had a while ago about why objects will reach thermal equilibrium even though they might have different specific heat capacities. To me, this didn't make sense as a body of water of the same mass and temperature will have more/less energy than any other material of the same mass and temperature. So why do they have different amounts of energy despite being in supposed equilibrium? I then considered temperature to be more of a rate of energy flow between objects. Similar to how a reversible chemical reaction will not necessarily reach equilibrium when there are equal amounts of the products and reactants, but rather it will reach equilibrium when the forward and back reactions are going at the same rate. This satisfied my curiosity as it explained why object holding different amounts of energy would be the same temperature and why neither would gain or lose energy if they were in contact with eachother.
But then I saw a veritasium video about how metals and plastics of the same temperature will melt ice at varying rates. If the metal and plastic were placed into contact there would be no macroscopic change as they are both the same temp, but if they were placed into contact with different pieces of ice (with each piece of ice being the equal in mass and temperature to eachother) they would transfer energy at different rates. This means I got my definition of temperature wrong, so I started to wonder what exactly is temperature? Information online states that it is a measure of the average kinetic energy or average speed of the molecules in a substance. But the weird thing is, telling me that something is 20 degrees tells me nothing about the average kinetic energy of the particles in it. What do degrees even mean? Speed for cars is measured in km/h but for temperature its just degrees.
 A: Your definition of temperature is essentially correct. If you came upon this definition on your own, I congratulate you, because it is much deeper and much closer to the fundamental statistical definition of temperature than the usual superficial statements about average kinetic energy that you will usually hear.
The following statement, which I'm paraphrasing from you, is a reasonably good definition of temperature:

Temperature is the property of an object that is the same for two objects that are in contact with each other if those two objects maintain no net exchange of energy between them.

This definition is a bit indirect. It doesn't tell you how to actually calculate an object's temperature, for example. But it's an accurate definition.
Now what about the issue with the video? (By the way, you haven't shared a link to the video so I have to go based on your description. If you share a link and it turns out I misunderstood, I'll edit/delete this answer.) The ice is colder than the metal/plastic, so in both cases energy is transferred to the ice. But you say that one transfers the energy faster than the other. This is not actually a problem for your understanding! It sounds like you're thinking, "Metal transfers energy faster than plastic even though they are at the same temperature, so the net flow between them would still be non-zero." (Let me know if I got you wrong.) But that's not quite right.
The only issue here is that you're comparing different pairings of the objects. The rate at which a substance transfers energy is not a universal property of that material. It depends on the situation and will change throughout the process of coming into equilibrium. The metal and plastic both at the same temperature do transfer energy to each other at the same rate if they are in contact. But metal at a warm temperature transfers energy to ice at a cold temperature at a different rate than plastic does to ice at the same two temperatures. Nevertheless
as the metal transfers more and more energy to the ice (or water if it melts), it will become progressively less willing to give more energy, and the ice will try progressively harder to give energy back to the metal until they reach a point when they are both equally willing to give/take energy. Then the net flow will become zero, the situation will reach equilibrium, and we would say they have the same temperature. The same will happen for the plastic and ice, although the details of how willing the plastic is to give energy at the beginning may be different, so that the melting begins more slowly. But in the end the plastic and ice/water will again "negotiate an agreement" and reach net zero flow.
This way of thinking about temperature is related to a concept called entropy, which very roughly is (the log of) the number of different microscopic configurations a system can be in with a given amount of energy. One of the principles of statistical mechanics is that systems like to maximize the number of different microscopic configurations that are accessible to them. It's usually the case that as an object's internal energy increases, the number configurations it can access increases. In your metal/ice scenario, initially the ice could dramatically increase the number of possible configurations if it had a little more energy. This in fact is what it means to be cold. Meanwhile, the metal's set of configurations would only decrease a relatively small amount if it lost energy. This again is what it means to be warm. So, fine, the metal gives a little energy to the ice, the ice opens up a lot more configurations, and the metal only loses a little. But as the metal gives more and more energy, it starts to lose more and more possible configurations for each chunck of energy it gives up. Eventually, it is no longer willing to do so, when the "benefit" to the ice equals the "harm" to the metal. The plastic and ice do the same thing, but the plastic, due to the details of its internal structure, may be more prone to losing configurations as it gives away energy, so it was less willing to do so from the very beginning, and the melting proceeded more slowly.
How does this all connect to the common statement about kinetic energy? If the particles that a system is made of can move around in some way, then as a system gains energy, the particles can move around more, roughly speaking, and that typically translates to more possible microscopic configurations. But you see that the explanation given above is much more fundamental, because it emerges from just considering any kind of configurations that the system have, which may not just be motion. For example, the Ising model is a model of magnetic materials that has all the properties such as temperature and phase transitions that you are familiar with, but nothing to do with kinetic energy. Your definition of temperature covers systems like this equally well.
If you're still not tired of reading all this, you can learn more about these topics from Schroeder's very nice book, which is also apparently summarized in this video.
A: 
So why do they have different amounts of energy despite being in
supposed equilibrium?

The short answer is because thermal equilibrium does not necessarily mean having the same internal energy.

I then considered temperature to be more of a rate of energy flow
between objects.

You are confusing temperature with heat. Temperature is a measure of the average random translational kinetic energy of the atoms and molecules at the microscopic level. Heat is energy transfer between objects due solely to temperature difference. So while a temperature difference between objects can result in the flow of energy between the objects in the form of heat, temperature is not itself the flow of energy.

Similar to how a reversible chemical reaction will not necessarily
reach equilibrium when there are equal amounts of the products and
reactants, but rather it will reach equilibrium when the forward and
back reactions are going at the same rate.

In a sense that is what is happening when thermal equilibrium occurs between two objects. When the temperatures of objects A and B are the same, the flow of energy from A to B equals the flow of energy from B to A for a net flow of zero. But that doesn't mean the internal energy of objects A and B are the same. For that you also need to know their temperatures and heat capacities.

This satisfied my curiosity as it explained why object holding
different amounts of energy would be the same temperature and why
neither would gain or lose energy if they were in contact with each
other.

And you would be essentially correct.

But then I saw a veritasium video about how metals and plastics of the
same temperature will melt ice at varying rates.

Yes, but that's only because the thermal conductivity of metal is greater than plastic. The rate of heat transfer depends both on temperature and the thermal conductivity. The amount of available energy depends on temperature and heat capacity.

Information online states that it is a measure of the average kinetic
energy or average speed of the molecules in a substance.

Yes, temperature is a measure of the average kinetic energy of the molecules (to be precise, we are talking about the average translational KE, not rotational or vibrational KE).

But the weird thing is, telling me that something is 20 degrees tells
me nothing about the average kinetic energy of the particles in it.
What do degrees even mean?

The temperature of something can tell you about the average  translational kinetic energy of the  particles. For example, in the case of an ideal gas, where $T$ is the kinetic temperature (K) and $k$ is the Boltzmann constant
$$KE_{avg}=\overline{\frac{mv^2}{2}}=\frac{3kT}{2}$$
Hope this helps.
A: In all cases, the temperature of a substance is determined by the average kinetic energy of its constituent atoms or molecules. Taking a container of water as an example, we note that the molecules in it have a varying range of individual kinetic energies because each of them has a slightly different velocity. Kinetic theory allows you to solve for that range of velocities, and the associated kinetic energies, but for convenience we associate this thing called "temperature" with the mean (i.e., most common value) of the kinetic energy distribution.
If we add heat to that container of water, we are making the molecules move faster, and the peak of the kinetic energy distribution (which defines the mean of the distribution) then shifts to a higher value- and we say its temperature has increased.
We can then set up a scale that divides up the temperature changes we observe into individual units, and call each of those units of temperature one degree.
A: Temperature is a value that describes the average kinetic energy of the molecules or atoms in an object. Imagine two gases, say hydrogen and carbon dioxide. Hydrogen molecules will be two atoms and an atomic weight of 2. The CO2 will be 44. In order for their temperatures to be equal, their average kinetic energies must be equal. This means the CO2 must have a much lower average velocity. And in collisions, momentum is transferred as if they are no different.
Take a block of tungsten and one of aluminum. Atomic weights 184 and 30. At the same temperature and pressed together, momentum at the interface is transferred both ways at the same rate. Why does one melt more ice than the other? Since they are not gasses you have to look for other explanations than just the definition of temperature. The density of tungsten is 19.3 g/cc and Al is 2.7. If the Aluminum block has 7 times the mass of the tungsten block would they melt the same amount of ice?
