The first suggests that when you say you've 'drawn a random variable from a population', your variable represents a single member of that population, randomly chosen. The second, and other definitions of the CLT or E[X] for example, suggest that a random variable drawn from a population is a random sample sample whose members are randomly chosen.

When someone talks of taking a 'random variable' from a population, is the exact meaning of what the r.v. is flexible, and depends on the context (e.g. on the particular formula) the variable is used in? Or is it more fixed? It's just been confusing me because as I try to understand the meaning of E[X], the CLT etc, I keep getting confused as to what from a statistical population each formula applies to.

Many thanks indeed, really appreciate your help.