The choice of subject and level influences the answer, I think.
If you want to learn general topology [as in a thread in meta-MO, I claim this is the same thing as point-set topology but sounds less old-fashioned] from scratch, then yes, I think it is preferable to pick up a good book -- e.g. Munkres, Kelley, Willard [not Bourbaki, IMHO] -- and work steadily through it.
However, if you go farther in general topology, it does become beneficial to compare different sources. (I think the word "passively" in Case 2 above is put there to make this case sound bad. Comparing different treatments of the same subject and trying to figure out whether they are really different is a quite active process.) I myself decided a couple of years ago that I wanted to revisit general topology (which I hadn't thought about since I was a 19 year-old undergraduate), and it has been very helpful to me to compare different sources. For instance, in my study of convergence I was quite baffled by the fact that any one book I looked at took a side on "nets versus filters" and then vaguely indicated that whichever one they didn't choose resulted in an equivalent theory. Only by comparing several different sources (and some research articles) was I able to figure out what was going on to my satisfaction: see Section 6 of
for what I learned.
For a different subject, flipping around might be a better approach from the start. Indeed you might not have a choice: as you go on in your study of mathematics, you find that it is very often the case that there is no unique text that is squarely focused on what you want to know (the bright side of this is that it is very exciting when a text comes out serving this purpose whereas previously there was none, e.g. Silverman's Arithmetic of Elliptic Curves).
Of course I agree that to really learn something you have to spend some time exploring it linearly. E.g., in order to internalize (even moderately) complicated definitions, you need to work out some proofs in which these definitions appear. Flipping around for comparison is not going to help you if you don't already have some sense of what you're reading.