There are some subtleties to physics not really presented in most lay publications. First, relativity was not invented by Einstein. The idea that physics (at least some aspects of physics) was not dependent upon the coordinate system used to display physical phenomena was first put forth by Galileo. Newton was the guy who really gave significance to the concept; however, at that time, it was presumed that the only reasonable coordinate system was the old Euclidean geometry. That is to say, that the "equations of relativity" (the relationships used to convert from one geometry to another) were a quite straight forward mechanical procedures. There was no "theory of relativity" as Euclidean relativity was held to be a fact and not a theory.
Newton's laws (as stated) were valid in an inertial coordinate system. If one were not in an inertial coordinate system, then one could use the Euclidean relativistic transformations to obtain those laws as they should be stated in a non inertial coordinate system. It is these transformations which give rise to what are often called pseudo forces. When free objects follow curved lines in your coordinate system, under Newton's laws they must be experiencing a force. There are two equivalent ways of viewing this circumstance. You can either say that your "true force" measurements must be made in the inertial frame or one can say that the use of a non-inertial frame produces a force.
In the second case (since the force is entirely due to the chosen coordinate system) that created force is generally referred to as a pseudo force. In many cases, it is actually more convenient to work in a non-inertial coordinate system. In particular, if most objects of interest remain at rest in that non-inertial frame (the earth, the interior of you car, whatever) descriptions of object behavior in the non-inertial frame become useful. Thus we have come to give names to some of these pseudo-forces. Centrifugal force is one and the coriolis force (very important in weather) is another.
Now pseudo-forces have one very important characteristic. Since they are entirely due to the coordinate system (if the coordinate system is not changing, the force vanishes) they can have absolutely nothing to do with any characteristics of the "free object" being observed. If the acceleration is entirely determined by the motion of the coordinate system (as the acceleration is actually a characteristic of the coordinate system being used all objects must display exactly the same acceleration), and the motion is to satisfy Newton's law, F=ma, the pseudo-force clearly must be directly proportional to mass.
Now two very important things came together around the turn of the ninetieth century. First, there were the problems created by the Michelson experiments. Maxwell's equations insisted that the speed of light was not a function of the coordinate system which was taken at the time to indicate that there must exist a unique coordinate system: i.e., that those equations only worked because our absolute velocity was slow as compared to the velocity of light (if we were moving, one would have to make the, very small, corrections required by the Euclidean relativity). Michelson was essentially intending to use those "required" small corrections to determine the exact speed of the earth through space.
We all know what Michelson discovered: the earth was standing absolutely still! Now this kind of blew a lot of physics away. Lorentz was able to come up with a set of relativistic transformations which would take care of the problem but could give no logical reason why they were necessary.
The other issue which had bothered some physicists very much was the fact that gravitational forces seemed to be directly proportional to mass. That is a very strong indicator that gravitational forces are actually pseudo-forces. The problem was, how could one change the coordinate system in order to remove the gravitational forces (i.e., get back to a real inertial system). In Newton's time, the only solution was to come up with a "real" force, but over the years, with the advances in non- Euclidean geometry, it was always hoped that someone might someday realize a transformation which would make gravity a pseudo-force.
At this point it should be evident to you why special relativity is called "special" relativity. Euclidean relativity was general (it told you how to make the relativistic transformations for absolutely any motion of your coordinate system). Einstein's theory of "special" relativity was an explanation of Lorenz's relativistic transformation equations based on the idea that the universe was not described by Euclidean geometry but was instead described by Minkowski geometry. On the other hand, the general transformations implied by Minkowski geometry just didn't work so there had to be an error in the idea somewhere. The solution eventually proposed by Einstein was a variation based on the "space-time" concept first suggested by Minkowski geometry.
I won't even talk about it here as "simple" just isn't an adjective generally applied to Einstein's general theory. In fact, very few physicists even attempt to express any common experiments in "General Relativisticly" correct terms. The actual achievement which makes Einstein's theory so overpowering is that it solved that old problem with gravity. He was able to construct a transformation which made gravity a pseudo-force.
Einstein's success led to the idea "that all physical phenomena must be independent of the coordinate system used to represent the phenomena". This is commonly expressed by the requirement that all phenomena should properly be expressed in what is called "covariant" form. What I was getting at with all this bull is that the idea of an "inertial" coordinate system is, very much, a Newtonian concept.
With regard to your second comment, yes, it is very clear that "time can be reasonably measured in inertial frames of reference". In that case, proper time (Einstein's path length in his geometry) and time (the coordinate in his geometry) are proportional and it makes no difference (mechanically) which you use. And, yes, "the measurement of time becomes more complex in accelerated frames"; however, the path length followed by your "clock" is a truly trivial thing to measure in that same accelerated frame. It is always exactly proportional to the reading on the clock. Doesn't that seem just a little strange to you?
Finally, I am willing to help anyone understand anything that I understand. Ask me a question and I will give you my best answer. Take a look at:
Have fun -- Dick