Thank you for posting. Personally, I believe the base system and bragging about it to be crude. Usually it's a way for (mostly) men to objectify (mostly) women without their consent. It turns intimacy into a contest, and that just really doesn't sit well with me. Neither is it an easy way to convey things. Yes, it's referring to physical sexual interactions, but sex looks different for everyone, especially those in non-heterosexual relationships. Essentially, it's really not the best way of measuring "progress" in intimacy.
Now before I answer your question, I want to say please take it with a grain of salt based on what I said above.
I believe it to be the following:
First base: Kissing, making out, etc.
Second base: Topless interactions
Third base: Bottomless/fully nude
Home run: Intercourse
Of course a "Home run" is confusing because oral, anal, and vaginal sex are all forms of intercourse. Some people may choose to include oral sex in third base, but that doesn't sit well because it makes it seem like the only "valid" form of intercourse is vaginal or anal, which is not true.
But basically, yeah, that's what they're talking about. I know I inserted a lot of personal opinion and I am sorry if that digressed from the answer, but I like educating people when they ask about things that are inherently misogynistic, homophobic/transphobic, etc (though that doesn't mean you are those things because you asked about them). I think it's good to spread awareness of that sort of thing.
PM me if you have any other questions. I am happy to help.