Multiple Naive Timelines: Given that the adult-you in the story is making the past materialize that he or she knows did materialize, I’d say that story is nicely set up to have a single static timeline. But neither adult-you nor the writer seems to realize that is even a possibility given the line: In some alternate universe, Malibu's prognosis must remain grim beyond this date.
Time Travel Methods
Time Booths, Wardrobes, et al.: Although not described as a booth, it is claustorphobically confined for the traveler and the cat together.