Is Einstein's characterization of "time" as "the position of the little hand of my watch" definitive and binding in the RT? In Einstein's very first publication dealing with the Theory of Relativity, effectively as a preamble to all subsequent thought-experimental considerations and descriptions, Einsteín put the suggestion
"[... that instead] of ``time´´ we substitute ``the position of the little hand of my watch´´."  [Punctuation marks as in the German original: Ann. Phys. 17, 891 (1905)]
Is this the definitive and binding characterization of "time" within the RT?,   
arguably together with (or based on) the understanding that any particular participant (such as "me", or "you", or "your watch" etc.) remains identifiable (as "the same" participant) even for distinct indications.
Is this characterization of "time" consistent or reconcilible with the notions "space-time" or "space-time continuum" which appeared (later) in certain other foundational literature on the RT ?
 A: Einstein says this, not because your watch is some ancient artifact with power over time, but because in the theory of relativity, time is relative. We can not longer say that the time is blah blah o clock everywhere. The time is different at different points. Therefore, your time is the time on your clock, and this is the "correct" time for your reference frame.
A: Answering this requires a bit of a preamble, so bear with me ...
Any observer can construct a coordinate system to locate points in spacetime. By construct a coordinate system I mean they choose a ruler for measuring distance and a clock for measuring time, then they choose the directions of their three spatial dimensions.
Even before Einstein, coordinate systems were not universal because your coordinate system needn't necessarily match mine. For example you might be using different rulers (miles instead of kilometres) and your axes might point in different directions to mine. But matching up our spatial axes would be pretty trivial, and both of us would agree on the time dimension.
What was new with Einstein's theory of special relativity is that the split between spatial and time dimensions was observer dependant. Both you and I could construct coordinate systems with three spatial dimensions and one time dimension, but if we are in relative motion then my time dimension wouldn't match yours. More precisely, my time dimension would consist of a bit of your time dimension plus a bit of your spatial dimensions, and likewise your time dimension would contain a bit of my spatial dimensions. The effect of our relative motion has been to mix up space and time. The result is that there is no longer a universally agreed time - each observer will now disagree on what is meant by time.
But there is a way out of this. Special relativity introduced a concept called proper time, $\tau$, that is defined by the equation:
$$ c^2d\tau^2 = c^2dt^2 - dx^2 - dy^2 - dz^2 \tag{1} $$
Let me explain what this means.
Suppose I have constructed a coordinate system $(t, x, y, z)$ that I'm using for locating points in spacetime. Suppose I'm watching you and I see that in an infinitesimal time $dt$ you move a distance $dx$ in the $x$ direction, $dy$ in the $y$ direction and $dz$ in the $z$ direction. If I feed the $dt$, $dx$, $dy$ and $dz$ into equation (1) it will calculate the proper time associated with your movement.
The crucial thing about this is that the proper time $d\tau$ is an invariant i.e. all observers using any coordinate systems will agree on the value of $d\tau$. I can't emphasise enough how important this is - all the weird stuff in SR like time dilation and length contraction can be derived from the simple principle that $d\tau$ is invariant. This single equation tells you everything you need to know to understand special relativity.
OK that's the end of the preamble, and we are now in a position to answer your question.
Suppose I use equation (1) to calculate my own proper time. In my own coordinate frame I am not moving so $dx = dy = dz = 0$, and that means equation (1) simplifies to:
$$ d\tau^2 = dt^2 $$
In other words, in my own frame proper time is the same as coordinate time - the proper time is just the time shown by my watch.
And this is Einstein's point. Different observers will disagree about what the time is, but all observers in all frames will agree that my proper time is the time shown by the position of the little hand of my watch.
This is also true in general relativity. What happens in GR is that the equation (1) becomes more complicated. For example near a black hole equation (1) becomes:
$$ c^2d\tau^2 = (1-\frac {2GM}{c^2r})c^2dt^2 - \frac1{1-\frac{2GM}{c^2r}} dr^2 - r^2 d\theta^2 - r^2\sin^2 \theta d\phi^2 $$
But the same principle applies. The proper time $d\tau$ is still invariant.
