what should I do?

Have sex with her, it's what you both want.

I doubt she wants me to fuck her.

Then why hang out?

Get sexual with her, do you want to remain frustrated?

Well, because I believe that, in her head, we can't be more than friends. And I also believe that she doesn't want to pursue a relationship with me. So we hang out as friends, like it has always been in the last 3 years.

But I'm not needy, nor desperate, so I walked away, because she blew me off.

You creating this topic is hardly you walking away. You didn't walk away.
Look. Listen to Heywood. Either get sexual, or stop seeing her all together.

She blew you off verbally. Because you chose to talk instead of act. This shit isn't something you just verbally agree upon, sign a deal, and bam, you're now fucking.

It's called seduction for a reason. So shut the fuck up and use your mouth in more creative ways.

I only did it this time, because I have a girlfriend and I wouldn't want to feel like I was betraying her.

The last two times we were together it was my invitation (though the last one I only suggested this week and told her I could only do it at 3 pm last thursday. I initiated our 3 last conversations, so I want to know the best course of action. I believe it's better to let her initiate the next conversation and invite me to hang out.

Your next thread on this same girl will be "HELP How do I get out of the Friend Zone!"

You have the attraction, but you have spent way to much time in 'comfort' being her #1 gal pal.

If a man spends too much time in comfort, he’ll get stuck in the friend zone. This usually happens when we don’t show/express sexual interest in her. When we don’t, often it’s because we don’t want to offend her in some way. like your betrayal excuse you make up. We won’t “offend” a woman by displaying sexual interest, if fact the opposite is true, it's a complement.

THE Complement.

But now you've been stuck in that cozy snugly warm comfortable friend zone sooooo long, that you are afraid of losing her if you dare step out.

So, long in fact you are clueless how you even would.

Well there you have it, your dilemma.

Now that you know, realize there is no reward without risk. The reward of your penis in her vagina. You remember it's the reason you posted this.

I will do nothing by now, only when she reinitiates contact with me.

And then?

showing sexual interest and touching her more.