> but to yourself, you don’t see any change
This hints to me that you might be thinking of the concept of invariant vs relativistic mass[0]
> i.e some sort of integral of moving matters between a distance, with respect to time.
It might be easier if you intuit it as total energy (encapsulates the relativisic aspects) - but that description you wrote seems to be another way to rephrase the curvature of space-time analogy; you could say that what you are describing is retreading how "steep" the bend is, yes?