You're right, Quartz, phase-in and phase-out both imply a long time frame.
Phasing out can take quite a bit of time, both among zoos and within a zoo. You may have to wait for all of the animals to die ("attrition"), or search for new homes for every one. But by putting the label "phase-out" on a species, it gets people in the mindset of seeing them go eventually ... it also means that breeding gets stopped (so that you don't increase the number of animals you have to get rid of).
"Phase-in" is a term that is usually only used by the TAGs (Taxon Advisory Groups) for a new species being brought to, say, North America. Because no zoos have them, there is a bit of a campaign that goes on to encourage institutions to accept/acquire this new species, and this can take some time (especially if a zoo has to phase out another species to make room!). When a specific zoo gets in a new species (e.g., when the red river hogs arrived in Toronto), it is usually labelled as an "acquisition" - you either have them, or you don't, so there isn't much of a time frame involved! (I should have probably used "acquired" instead of "phased in" in my message above)