Yes, I am aware of those points.
Point four is, of course, one of the fundamentals of the theory of relativity.
It equates mass and energy. But can it be reversed to infer a change in mass resulting from a change in electrical charge?
I mean physical mass rather than relativistic mass?
Yes. It is a matter of the system definition and reference frame.
By physical mass, you mean rest mass. But even that requires a closer look as it is not as different as you might think. Let's look through our universal scope to see what I mean.
To start, keep it simple and say and object's mass is a property that can be measured by looking at an the object's acceleration in response to an applied force (inertial mass). We could also look at how the object responds to a gravitational field (gravitational mass). While the inertial mass and gravitational mass can be measured multiple ways, it is accepted that these are equivalent (the Equivalence Principle). They have been measured over the years and they keep finding them to be equivalent (currently have been found to be closer than one part in 10^13).
Let's look through our scope at a far off view. Look at the response of one stellar body to another's gravitational pull, maybe a star. We can measure this response and calculate a mass for that body. If we zoom into that body, we find that it is actually not one physical body but is made up of other physical bodies, energies, etc. Great. Now we know that a stellar body can be considered as one mass or a collection of masses. But what about real life here on earth?
When we measure the rest mass of an object, our reference frame is such that the object as a whole does not seem to move. This is your "physical mass". Put that body in motion relative to the reference frame and it gains additional mass. This is your "relativistic mass". So why not just measure everything standing still?
We can't measure a photon standing still because it moves at the speed of light. It's mass is made up of "pure" energy. If we attach our reference frame to the photon, our frame is now moving at the speed of light and the photon will have zero mass (i.e. it will have no rest mass). The other question is what does "standing still" mean?
Consider an object at rest that we can see with our eyes. We can measure its rest mass. But that is not the complete story. That is the rest mass given a defined system and reference frame. Our system boundary is an infinitely thin shell on the surface of the object and our reference frame is the center of the object. Consider what happens when we zoom in on this object with our scope.
As we focus in a little closer, we begin to see smaller objects: molecules, atoms, electrons zipping around, maybe even some light or heat waves, etc. We can even see some bonds between atoms that are stretched because of an elevated energy state. So rest mass is relative itself. What we called rest mass when we were zoomed out is now revealed to actually be a combination of even smaller "rest masses" and other energies. In a previous frame we called all of this one system. Now we see a whole group of smaller systems.
What happens when we zoom in even closer on the smallest of these small systems? Some think that we will just find some energy in a bound state. Nobody knows. But as far as we know, it is all just energy in some form. "Rest mass" is just another form of energy. What we decide to be "rest mass" depends on one's reference frame.
Physicists change reference frames when zooming in like other folks change shirts. They can be so buried in relative frames that it looks like a nightmare, but that is their playground. Then when they talk to us normal folks, they can confuse the stew out of us because we don't live and breath the same assumptions that they take for granted and they just figure you should know what they mean.
So what do we now realize? All those little bits of energy inside a system do make up the total mass of the system. We can define some of that energy as "rest mass" and some as "relativistic mass" but it is all part of the system's total mass. It just depends on what you are focusing on.
So how do we get mass from a "pure energy" object like a photon? Enclose it within a system, then zoom out, and it is now part of the system mass, just like some of the photons that are zipping around inside the star we looked at from the very beginning.
What happens when an object absorbs energy? It becomes part of the system mass until it is released again. Heat up the center of a rock at rest and the rock will gain rest mass. The energy is now enclosed within the system boundary defined by an infinitely thin skin on the surface of the rock.
Enclose a twisted rubber band by a system boundary and the energy that is stored in those stretched molecular bonds now adds to the system mass. If the rubber band unwinds, it will create kinetic energy, heat energy, etc. that can escape our system boundary and cause our rubber band system to lose mass.
What about a rubber band that is "stuck"? Let's call it a molecule instead of a rubber band. This molecule has a rest mass m1. Find a object with rest mass m2. Let's define a system enclosing the molecule and object. The system has rest mass m1+m2 = m.
We have found that we can smash the object into the molecule and get two new objects with mass m' = m1'+m2' < m . All momentums are accounted for to find the new rest mass m'. We measure a quantity of energy released that measures E. We find that it is exactly equal to the reduction in system mass by E = (m - m')*c^2. This has been done over and over through the years in many labs and it continues to verify the rest mass-energy equation of E=mc^2.