In general, when I say, "clocks" run slower... what I mean is that time flows at a different rate.
Yeah, relative to someone in a "static" frame of reference. But ten seconds flows at exactly the same rate from your pov regardless of the velocity you're travelling at. The slower clock thing, it's other people who see your clock going slower. Time always flows at the same rate from your pov, a second is always a second.

That's my understanding anyway. Please feel free to tell me I'm wrong, I'll believe you!