I think the best thing to keep in mind is people’s level of knowledge and attitude regarding the Jedi and the Force is probably very different relative to an individual’s age, social class, home world, etc. in the Star Wars galaxy.
Some people might not like the Empire, and be old enough to remember the legendary stories of the Jedi, so they could still hold reverence for them, like Lor San Tekka. Others might be young and brainwashed by the Empire and think the Jedi are dangerous and evil, like Del Meeko from Inferno Squad.
I think generally, after the initial time around the Jedi Purge and labeling them as traitors once the Jedi were almost completely wiped out, the Empire began a massive campaign to censor history and wipe the Jedi and the Force from the collective conscious of the galaxy. For example, in novel Lost Stars, recruits in Imperial Academy were educated that the Clone Wars were sparked by extremists interfering in a legal execution, not even mentioning the term “Jedi”. It seems some Imperials learned that Jedi were potential dangerous targets, but they likely didn’t know the context of their existence beyond them being religious extremists.
Like I said, the knowledge of common Galactic citizens was extremely varied, but I think for people who grew up in the Empire, like Luke, were likely not educated about the Jedi or the Force at all, unless family or friends shared stories about them. So Obi-Wan probably wasn’t worried about telling Luke he was a Jedi, because the Empire didn’t want to even teach the youth about the “traitorous” Jedi. They wanted to erase them from history.
Specifically for Luke, it is possible Obi-Wan knew he didn’t like the Empire, or he knew it is unlikely Owen and Beru would want to teach Luke about the Jedi at all, in fear of Luke’s curiosity leading him down a path they didn’t want. Also, clearly Obi-Wan wanted Luke to learn the ways of the Force, and would probably be willing to take the risk of revealing his identity as a Jedi in order to re-educate him, if it was a risk at all.
In regards to Tarkin knowing who Vader is, the canon novel Tarkin makes it pretty clear that he knows Vader is Anakin Skywalker. I believe he deduced that since he knew Anakin during the Clone Wars and could compare the two individuals enough to recognize them as one in the same. I imagine at some point, Vader realized Tarkin knew, and afterwards wasn’t concerned with trying to hide it from him. But I believe was one of the few people who knew this fact.
This is just my understanding, but I hope this was helpful!