Or to put it another way, what you are saying (I think) is that if Joe and I had two absolutely identical clocks, and he was on Everest while I at sea level, his would run faster. But is that really time dilation, or just because the mechanical process of his clock is in some way affected by less gravity ? To draw another, most probably naive analogy, I could get water vapour in my clock, and that could slow the mechanical process down substantially in comparison to his. That's not time dilation, is it ?
First, there is always some risk of when attempting to describe these things in everyday terminology. And even more when attempting to break it down into components, as I have been... Keep in mind also that the only way "I" can try to describe this is heavily influenced by a philosophical perspective.
I sometimes think of SR as establishing two major concepts.
One is that reality is completely observer dependent and how we can from what we see project that, to an understanding of what someone else would see. This part is covered well within the context of the simultaneity or relativity, which you seem to understand. This also extends to the understanding that the rules or laws of physics are the same for all inertial observers. Observer's who may be moving relative to one another but are not accelerating, relative to one another.
The other, deals more with an attempt to describe the world as it exists, beyond those subjective experiences. What is the underlying truth of the world, that we see from where we each stand. The problem is we can only reach any global or underlying understanding from those subjective perspectives.
So, on to time dilation, which is the essence of your question.
Trying to define what time "is", beyond our subjective experience, is a subject that may never have an entirely complete and satisfying conclusion. All we can really say with certainty is that time is, for us.., an observation and measurement of change. Understanding this it does appear from experience that the rate of change of anything we are able to observe is affected by where in a gravitational field it is located. In your example identical clocks at sea level and on top of Mt. Everest. You add the question, is this really time dilation or maybe just that the mechanism of the clock works differently in those two locations. The answer is yes, to both!
Remember, for us to associate time with a clock is a subjective abstraction. Where the physics is concerned it is really not how our experience of time might change in different locations, it is how the change represented by that subjective observation does change, under different conditions. And though there are many things that can and do affect the rate of clocks, time dilation is specifically associated with velocity (SR) and gravity and/or accelleration (GR). Mostly because these are conditions that define our frame of reference with the context of physics.
So to return to your question, "
... But is that really time dilation, or just because the mechanical process of his clock is in some way affected by less gravity ?...". Again, yes and yes. Think here that, "today" we use very accurate atomic clocks and they are affected by where in a gravitational field they are. "Time dilation" is nothing more than observing and at this time.., today.., knowing that change occurs predictably at different rates depending on where in a gravitational field you are located. Time is just the way we measure the rate of change.
The fact that even the most fundamental changes, those associated with atomic functions, both those utilized in atomic clocks and the apparent half lives of subatomic particles, all seem to be affected predictably by velocity (SR) and gravity and/or accelleration (GR), each in their own way, demonstrates that it is real. The changes do happen and we call and measure those changes, as time.
Time dilation is a definition and description, of how the velocity or location within gravitational field, predictably affects, the rate of change, uniformly.
