In both situations I was the one to approach/ask out.

I've a friend who is really good at approaching in public. One of the things she does is ask for help taking a group picture and from there, she is able to start conversations.

From time to time, I can be social and talk to people. But in my case, is usually something nerdy, like commenting on a T-shirt. People who is interested, continue conversation. Then again, each person has their own likes and style.

When I talk about the times it has worked for me, I mean long term relationship (the second still on going). But if you want to count short term, maybe my rate is higher. XD

For example: with current boyfriend, I did noticed that his picture was at a historic place so when I opened conversation, I did asked if he liked history. and also about the board games he likes. From that, conversation was fluid. Same in person. Then again I could "pick up" clues from clothes, books, or details in the profiles of men. You can do that in person or at a bar. And also (more difficult to explain) you can pick up on body language.

So I guess my point is, people expect that there is a "pick up" method that works for everyone. But I think it is the opposite. Every person is different. You might be only compatible with a few, so what matters is to be able to identify the people you might be compatible and adapt/personalize the way you approach to the specific person you are approaching. They are unique and so are you.

Also, the sooner you identify the people that is not going to be right for you, the better. For example: someone sexist is never going to be right for me. I'm open about my feminism, but I also learned fast that many men lie about them not being sexist because they want sex or, worse, they want to "redeem" you from your feminist (basically train you as if you were a pet so you put up with their sit). So what you need to do is pay attention to what they do, regardless what they say.

For sexist people it should be the opposite. I recommended a friend that if he wanted a traditional wife, he should "signal" that in dates. And suck it up and pay for dates, open doors, etc. Sure, he didn't like paying. But a traditional women wants that.

Asking/approaching is just a first step. You might talk to them or go on a date and discover it won't work even if the date is somehow pleasant. For example: you might offer to pay and he throw a tantrum or react in a weird way. Or maybe he let it slip that he expects you to agree on everything (in fandoms this could be terrible XD). There is no need for drama, but you might want to accept that it won't work and not pursue anything more.

Lexie would do well because she is honest about who she is. And if she is also honest about what she wants in a partner, she would develop ways to identify that person by their actions. People lie, but they can't act all the time.