Any standard meaning of the term "bounded (stochastic) process"?

Question

I always get confused when an author refers to a "bounded process", because it seems like it can take two meanings depending on the author. For example, in Rogers and Williams, it seems to mean $X(t, \omega) \leq K $, $\forall (t, \omega) $, whereas in Daelban and Schachemeyer, it means $\forall \omega$, $X(t, \omega) \leq K(\omega) $, $\forall t$ instead, i.e. one means the paths are all uniformly bounded and the other means the paths are bounded but not necessarily uniformly so. The annoying thing is that both books don't even define the term, so I am left unsure.