The T-800 didn't really care about JohnWhen John Connor reprogrammed it, he just had it pretend to gain human emotions so that his past self would trust him.
Changes to history only ripple from when they started onwards.Hence why Connor still exists when there's no SKYNET to eventually motivate him to send Kyle back to father him, and why they even remember the events of the 2 Terminator movies when the future can't prevent them going back in the first place. While the present changing the past does change the present(or else SKYNET's plan was All for Nothing), the present changing the future doesn't change the present even if those events directly led to the present they know. It also solves the potential paradox of if the original T-800 or the T-1000 won; the future would see no reason to send them back, but as the ripple effect only goes forwards it wouldn't affect them. Terminator Genisys supports this with T-3000 John Connor being an "exile in time" and able to pull off a Grandfather Paradox, by virtue of being under the same circumstances as John Connor preventing SKYNET existing in this movie. This makes the T-1000 even more of a gamble because if it realizes this, it would be wholly willing to turn traitor and warp SKYNET's history to its own liking or just Ret-Gone it and be independent without risk.