Sure, Mountain Thrush might show up in San Jose at times, but keep in mind that many of the eBird records are likely recorder error (I looked through them and only saw one defensible record, the majority of the records from the city proper are from visiting birders or are just patently unlikely). If/when Mountain Thrush occurs in San Jose it is going to be many orders of magnitude less common than Clay-colored, and when it does occur, is likely to be in parks or wood lots, or in gardens with big trees, more than perched on hoses and fences, and is more likely to be at the edge of the city than in the middle of town. Anything is possible, but you're more likely to win a mega millions lottery than photograph three Mountain Thrushes out of a hotel window in San Jose
Within the city of San Jose there's only one record of a "Mountain Thrush" with a photo -
https://ebird.org/checklist/S102222643. I believe Mountain Thrush should always have dark marking on the undertail coverts which this bird does not have, but I do not know if Mountain Thrush can lack this field mark. The bird in this list looks like it could be Pale-vented Thrush to my eye. I would want to dig into ID of those and refresh myself a lot more before offering a stronger opinion.
There is one record that looks believable, the description is of a Mountain Thrush from someone who knew what to look for:
https://ebird.org/checklist/S41131754
As far as the original three pictures posted:
-The first bird looks to have a much warmer tone and perhaps pale legs, though I would weight the probability of occurrence far more than what we can see in the photo.
-The second bird probably has a pale bill though the photo is not helping. Again, I'd go with Occam's Razor here and say that unless you have a very good photo or recording of a definite Mountain Thrush, you are much safer to just call it a Clay-colored Thrush.
-The third bird has a yellow bill and enough detail is visible to feel quite safe, even from just the photo, calling it a Clay-colored Thrush.
If someone really expert on these species takes a looks at the photos and suggests one is a Mountain Thrush, I'm all ears and keen to learn, but I don't expect it to happen is all