You say that "our count stays the same". Compared to what? You also note that "the clock that moved is behind the other in its count".
The word, TIME, in the case cited above, refers to an index (the count, the numerical readout of the clock's accumulator) of the atomic transitions.
The word, TIME, in general, refers to indexes of incongruent spatial configurations.
Time indexes all have to do with the association of one set of spatial configurations with another.
The duration of a particular spatial configuration (or set thereof) is defined (measured) by associating the interval defined by it's appearance and disappearance with an index of some set of incongruent spatial configurations that are a subset of the aforementioned interval (usually the count of a 'regular' oscillator, like the one you mention or a vibrating crystal, etc.).
And I agree that there's no absolute measure of the duration of any particular spatial configuration.
There's probably a better way to say what I said. So, if you basically agree with it, or not, take a shot.