The perception of time for both is subjective.
The objective time is what they are both perceiving through their "lenses".
A reflection in a twisted mirror that creates a different distortion depending where you stand to look at it.
Each observer has a different picture but what is creating the picture is one thing.
I was tired, and I must have forgotten what my point was. So disregard my "Okay, I see" response.
When the clocks come back together, they will show different times. This is evidence that time dilation goes beyond just perception and subjectivity. Most importantly, it shows that one event, the mechanical process of the clock in space, happened before the identical process of the clock on Earth.
A better example is when two identical clocks start at the same time. Then if one clock enters a stronger gravitational field, it will objectively run slower than the other clock.