[Gorjus]

Are they only speaking of the portion of the Universe that we can either see or measure from Earth or by other means?

Kinda. There's pretty good evidence that everything in the universe started from a single point (the 'Big Bang') and so the 'edge' of the universe is wherever the objects are that have gotten farthest from that point.

Whatever might be outside that is unknown. Perhaps there's an expanding wave front from some other Big Bang a zillion light years or so away, which will someday overlap with the expanding wavefront from our Big Bang. It may even have already happened. All we know is that nothing outside our own 'universe' has appeared.

Perhaps there truly is an infinite (in three dimensions, at least) empty space beyond the farthest material from our own Big Bang (though there are some strange observations that imply the universe closes in on itself so that traveling in a 'straight line' doesn't really take you off into infinity).

[Doctor Stochastic]

... the 'edge' of the universe is wherever the objects are that have gotten farthest from that point.

Not quite geometrically correct. There is no "that point"; all points are equally that. It's the whole thing that's expanding.