Why do we need Higgs field to re-explain mass, but not charge? We already had definition of mass based on gravitational interactions since before Higgs. It's similar to charge which is defined based on electromagnetic interactions of particles.
Why did Higgs need to introduce concept of universe-wide Higgs field to define mass, based on interactions with it? And nobody cared about the charge of an electron (for example), which is also basic attribute and constant?
 A: Once again, I am way out of my league in answering this. I may be wrong about many things here, comments appreciated
That was just a definition of mass. The Higgs explains where rest mass (but not gravity) comes from in a mathematically rigorous manner.
One of the attempts to explain how our universe works in a mathematically rigorous manner is the Standard Model. It tries to put all the fundamental forces (except gravity) under one umbrella, along with the particles in a comprehensive theory that explains the subatomic world in a consistent manner.
The theory, in the course of its evolution, has predicted many particles--including many of the quarks, the $W$ and $Z$ bosons, and of course the Higgs boson. All except the Higgs{*} have been experimentally confirmed. 
All of these particles are necessary for the theory to work. The Higgs is a product of a neat mathematical trick (spontaneous symmetry breaking), that leads to the concept of "mass" blossoming into existence. 
If the Higgs is not found, the whole theory does not work (or needs significant tweaking). IIRC, the SM initially predicted a massless $W$ particle and had inconsistencies--which were resolved by introducing spontaneous symmetry breaking. The original intention for introducing spontaneous symmetry breaking (and thus the Higgs), was to "fix" this issue in the electroweak interaction. From this point of view, the "Higgs makes particles massive" is more of a side-effect, an afterthought. 


Mass is not exactly comparable to charge. Mass has two aspects--the gravitational aspect and the inertia aspect. EM forces are the "charge" analogues of gravity, but there is no such analogue for inertia.
Now, the Higgs is explaining the inertia aspect of mass, of which there is no electric counterpart. So there is no need to assume the existence of an electric analogue of the Higgs.
As for EM forces, they are already explained by the SM (the force is mediated by photons). The SM specifically leaves out gravity, but there is a hypothetical particle, the graviton, being researched--this may explain gravity as well.

As @David has mentioned, the above section isn't exactly correct. THe reason we don't need another particle for charge is that charge is already explained by the SM, mathematically (whereas, without Higgs, rest mass is something we just assume)
*As of today, this has probably changed--a particle that is similar to the Standard Model Higgs has been discovered by CERN
A: Actually, mass and charge are only superficially similar. Yes, they both appear in inverse square force laws, namely Newton's law of gravitation and Coulomb's law of electrostatic force, but both of those are approximations. Coulomb's law ignores quantum effects, which is a very slight approximation, but Newton's law ignores all of relativity, which makes a huge difference under certain circumstances. The true underlying theories, quantum electrodynamics and general relativity, are almost completely different.
Now, to address your questions directly (though admittedly, this would be a lot easier to explain with the math):

Why did Higgs need to introduce concept of universe-wide Higgs field to define mass based on interactions with it? And, no body cared about charge of electron (for example) which is also basic attribute and constant?

Think about this: in either Newtonian gravity or general relativity, mass is a property that you just assume an object has. Neither of those theories makes any attempt to explain where the mass of an object comes from; the mass is just something you plug into the equation to calculate a trajectory or a force.
The standard model is more ambitious than that, though: it wants to actually explain things, not just have them put into the theory by hand. It all starts with a principle called local gauge invariance. By going through the math, we find that the consequences of this principle correspond to many of the same properties we know particles to have. For example, one consequence of local gauge invariance is the fact that some particles have electric charge, and the existence of the electromagnetic force. Another consequence is that particles have "color charge" which leads to the existence of the strong force. It predicts the existence of antiparticles and the correct conservation laws that govern which elementary particle reactions can and cannot occur in nature.
But before the Higgs mechanism was discovered, the one thing the standard model did not predict was mass. In fact, all the particles it predicted to exist, which in almost all other respects matched known particles exactly, would be massless! Sure, we could tweak the standard model to force the particles to have mass, but there was no particular reason to do that (other than the fact that we know the particles have mass in real life). There was no simple principle that would require the theory to include mass the way local gauge invariance requires the theory to include electric charge, color charge, etc.
What Higgs and other scientists (Anderson, Brout, Englert, Guralnik, Hagen, Higgs, and Kibble) discovered is that the principle of spontaneous symmetry breaking does exactly that: it enables, and in fact requires, the particles of the standard model to have mass. The neat thing is that it only does this in combination with local gauge invariance, but that's kind of beside the point here. The important thing is that when you add spontaneous symmetry breaking in to the standard model, you get particles with mass, where before you had massless particles. In order to add spontaneous symmetry breaking, you need to add a field whose symmetry can be broken. That's where the Higgs field comes from.
