[Ilya]

Something has caught my eye in your post:

Originally Posted by MartinT

No, getting laid is not my goal – the primary thing I would like for now is just a simple relationship with a girl. Even if it just means getting the 2nd date.

and this:

and finishing with me, thinking like “ah, she wasn’t interested in me at all. No, really. Why should she? No, no, it’s not a good idea to call her again. Let’s wait for tomorrow.”.

So, you are able to get a first date and something breaks down before the second one. And I'm afraid to say, the main problem here is you not calling back. If a girl went on a date with you once, she is clearly interested in you. Believe me, when they are not, they have a million ways to avoid the date.
So the girl is interested, you take her on a date, you have reasonably good time and... nothing. Despite your fears, the main problem is that you are being impolite. You must call even if just to thank her for her time. Otherwise you are being rude.

So not calling is really not an option. And since you must call, why not suggest another date at the same time. To make it a little be easier, try to set it up on the first date. Promise to give her some book, or an item, that is connected to the conversation. This will give you a legitimate pretext to set up a date. If she agrees to a second date - you have a relationship. Really. No one goes on two dates with a person they don't like. On the other hand if she starts to make lame excuses, then, it might be that you did something wrong on the first date. But make this conclusion if she avoids a second date for, say 3 times. One time, she can be busy, and try to work out the time that's acceptable to both of you. 2 times - strange, but stuff happens. 3 -times, she is avoiding you. Say goodbye, tell her she can call you when she feels like meeting (just in case) and move your attention to another girl.

Oh, believe me, I know the theory :-) Implementing these things in practice is the biggest problem… But, as also noted in that article, it's social/dating SKILLS - and they can only be learned by practice, not by theory and not by thinking. I must somehow get that into my head.

Well, true, but that's the part of the article, that should not be taken literally. You do both. Preparation does help. But it is useless without action.
For me, in the early days of dating, the main problem was that I didn't know the whole process. And I cut short the dates and intentionally scared the girls as soon as I entered uncharted waters. You know, it was the classic case of fearing the success. First date, and I'm already worried how am I going to propose to go to my place and what is going to happen afterwards. So I started reading a lot. I have to admit that I'm probably the only person in the world who learned to have sex from books. It is stupid, I know, but by the time I finally had a chance to use this knowledge, I've known my way around pretty and my first intimate partner never believed that it was my first time. So, sometimes it helps to be ready to go all the way and be ready to what can happen there.

Also, you seem to have a great asset in your social life. Humor. It helps a lot. You have a problem and you still can make jokes about it and not be too serious about it. That is a very good attitude to have in your situation. Keep it up.