Hmmm. I don’t think this is easy to determine because there is usually flirting or conversation first that indicates a mutual interest in each other. Maybe I make the first overt move (do you want to get a bite to eat?), but that’s based on some pretty intense interest that already exists.
Online, the other person has made the first move—if you count announcing their existence to me as a move. I mean, I don’t know they exist, necessarily, and then I get a pm which, if I read between the lines properly, is an expression of interest in me. I have written similar pms as well. They are subtle, with plausable deniability. You just say hello, and ask a question, and if they don’t answer, there’s nothing there and no harm done. No real reason to be hurt. But if they do answer, that opens up to the next step, which involves sharing more personal information, and then, step by step, you increase intimacy and if it works out, you end up… well…. where relationships end up. It’s hard to say who made the first move in terms of how did it get intense, but I do think that first pm is often the first move, and sometimes the first move occurs out in public, as well.
I think that men often don’t recognize when women make the first move. I know I didn’t for most of my life. Real confident men might respond to a woman’s first move, but it’s not clear to me that they recognize she is telling them she is available for them, if they want. Women can be subtle about that. Flirting has many ways and means, and I think guys tend to think they are more in control than they are, while women tend to think they are less in control than they are.
Women sit back, and let men make the “first move” very often. Except, I don’t think that official first move is even close to the first move, usually. But it’s just the first one related to dating—the first official “ask.”
I’m shy, so I tend to let her get fairly obvious about her interest before I’ll risk asking for something. I also don’t ask for dates. I make it more slippery than that—like a non-date, date. “Are you planning to be at such-and-such next week? A bunch of us are going. Cool. It would be great to see you there.” Well, it worked for me.