^^^ If it's a used guitar then there's a good likelyhood the pickups have simply been replaced. There are lots of Jazzmaster replacement pickups that have P-90s or humbuckers under the covers, with screw poles used in both either for authenticity (P-90) or just to fill the space easily and cheaply (humbuckers). In fact I would wager the best-selling JM replacements are either P-90s or HBs and not 'true' JM pickups. They certainly were what we ordered most of in the store here when I was on board, some years back. (The 2010 emo and pop-punk trends really drove that; Jazzmasters became hugely popular overnight, but not for their normal tone.)
Anyhoo.
Assuming they're not limited edition or custom hybrids and we just go with the standard 50s & 60s designs, I'll say a Jazzmaster is a must. It is more different in sound to a Tele or Strat than the other two are to each other and until you get up to the highest frets it's the most balanced and comfortable, too. (Fret access beyond 16 or so is worse, though.)
Strats and Teles do/can sound so close to each other that having both is a bit redundant, unless you're in a position where you can really 'optimise' them for different sounds, or use them for different tunings. A totally standard Strat tuned to E Standard and a totally standard Tele tuned E Standard are going to sound 90% the same through most amps and with most playing styles. Yes, if you A:B them directly next to each other on an isolated track you can hear an immediate difference, but that difference goes away once you factor in playing at volume, having a drummer and bass player going, maybe a second or third guitarist, a singer or two, keys, etc. Ultimately the Strat's defining sound is it gets a middle pickup (which many people ignore anyway) while the Tele's is that there is a little more tonal 'contrast' between its bridge and neck pickups than there is between the Strat's. I wouldn't factor in the vibrato since the Jazzmaster also has one, and we've established for variety the Jazzmaster is making the cut.
So for me it basically comes down to whether or not you'll actually use the middle pickup. Some love it, some never touch it. Personally, I use the middle pickup in all guitars more than I use any neck pickup. For that reason, if I had to only keep one and throw away the other style for good, I'd go with the Strat. If you're one of those people who never gives a second thought to middle pickups then the Tele will be the better bet, since it's that bit more different form the Jazzmaster. However I would also note that if you ever come to want to modify the guitars, it's usually easier to modify Stratocasters, and they can be modified to more extreme lengths, than Teles.
But, really, you could just flip a coin for Strat v Tele. Case in point, if you ever look up any advice or videos to sound like Prince you'll see many people telling you to get a Tele, since that's the shape of guitar he played... but his guitar had Strat pickups in it, mounted through pickguard material with no baseplates or anything, so it didn't actually sound like a Tele. It was much more like a Strat, just the shape of a Tele... and yet many people still use a Tele to imitate him and you know what, it does the job just fine. Similarly I've got a Mustang with some Tele pickups in it and guess what, it sounds like a Tele and you'd barely tell in a recording that it was a differently-shaped 24" guitar, and live there is absolutely zero audible difference. They're all that interchangeable in tone.
