Is gender based on men and women doing different things?
Sort of. Masculinity/femininity is literally just anything that's traditionally associated with the male or female love.
When somebody is talking about gender expression, they're talking about whether someone expresses themselves as more masculine or more feminine.
There's no 'correct' definition because both definitions developed organically and are used by large groups of people. Ironically, SJW communities tend to overlap well with linguist communities, so if you want to piss someone off on tumblr, call them a prescriptivist.
"Even though I have a richard you should call me 'she'"
The whole pronoun thing is the other complicated part of the debate. Nobody can force you to speak a certain way, but there's not a whole lot of good reasons to refer to someone by a pronoun that they don't want.
There's a pretty large body of evidence that people who identify as transgender have a totally automatic and subconscious response to being called the wrong pronoun. It's the same kind of feeling that you would experience if someone called you 'she'. Likewise, if someone called you a 'she', you'd probably feel inclined to point out that you're actually male. It's not something that they stop and think about just to bug you, it's pretty much completely automatic.
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