Yes, Lynbrook is better. But the linked house in San Jose does not go to Lynbrook.
To OP: I’d buy in best location possible with crappiest (or smallest) house you can stand. You said enough room for in-laws, so does that mean 3 bedrooms, 2 baths? $2.2M for 3/2 can get you into most cities in peninsula. If your requirement is 4/2 or 4/3 for $2.2M, then your choices are a bit limited but still many options. Not knowing any more info, cannot give any useful opinion or advice.
Just to throw it out there, how about Fremont? With a budget like that, you should be able to buy in Mission San Jose neighborhood with its great schools and you should get a pretty nice home with a large lot. Weather in Fremont is pretty nice and warm. Each of you would have to commute though by Bart to the city or drive across bridge to South Bay but doable I think. Sure, Fremont may not have the cache of some of the other places mentioned but it is still a desirable city where a lot of tech workers are moving into. In fact, maybe you can buy a smaller house and build/design your dream home there instead of having to settle for something already made.
What square footage are you looking for/how many bedrooms in the non-inlaw part? Sounds like PA might be the place to look so that you can split the commute.
Are you ok with Menlo Atherton as a high school? They have a mixed demographic with an honors program. If so, Menlo Park might also be a good match. Encinal and some of the other MP elementary schools are supposed to be good. Hillview is recently built and people I know with kids there are happy.
Our house hit the market today. It’s small but updated, sits on a lot that’s perfect for adding on. The assigned elementary isnt great but it’s easy enough to transfer to a better school that’s just as close. It’s super convenient for commutes.
Short drive to diridion station, even shorter drive to Santa Clara station.