An update on things. Tl;dr: is good!
It took nearly a month before I could take the second drive. The instructor was ill, but luckily it wasn't anything serious. However, with such a long break, my anxiety has returned. In a hindsight, it was a very good idea to pay for several lessons in advance: this way I had no choice but to go on. The instructor, being great as she is, made up for the break by making the lesson over 1.5 hour long. Including the refueling - they don't typically include this on regular driving licence courses. Sure this is a no-brainer once you know it, but what if you have never ever done that?... Also, the first drive was largely outside city environment - the second was right within it! I got awfully exhausted mentally and emotionally as a result. But I thought, maybe it works like physical training, it over-extended myself but should gain some strength as a result. And I did.
The third lesson was also in city traffic, but started including parking. Now, this actually was one of my greatest fears. I had to learn parking to get the licence, but my original instructor was kinda messy about that and I never got confident with parking, good that they didn't stress that aspect on my exam too much. But this instructor now, she showed it in such simple terms that it turned out to be stunningly easy, compared to how I remembered it. I feel that I'm somehow more fond of parking backward than forward now, whereas previously it was like an unthinkable stunt! This third lesson turned out to be a breakthrough, I was still somewhat anxious at the beginning, but when I saw that I can do even parking - it was like a permanent anxiety relief.
I'm after a fourth lesson now, which was much like the third one. But it was the first one done without any significant anxiety! It wasn't 100% perfect technically still, I manage to choke the engine on some occasions. But if I had a car with automatic transmission, I'd say I'm ready to drive on my own now.
With manual one, it will take a few lessons more to iron things out and get even more used to the whole thing. But overall, this whole plan is working out. I'm not in a good position to get a vehicle anytime really soon, but if/when time comes - I'll be ready.
And now, for a more "philosophical" observation. Maybe it will help someone. When seeing the traffic from the perspective of a pedestrian, or even a passenger, it all looks like freakin' huge colorful chaos. And that's how I perceived it for all these years. But when actually being right within it and driving with actual engagement and motivation - it struck me that there's an order in this mess. The road signs show what is allowed and what is not, i.e. what are your options. Traffic lights show when to go. The lanes are marked. If you go straight then you go straight, if you want to turn then you find the proper lane, or the place to turn into a side road is marked as well. Parking may require centimeter-level precision, but has a clear set of methods and is actually doable.
Taken together, there is rhyme and reason in driving, but frankly, it feels as if I needed certain level of maturity to recognize that. The right time had to come, and that time certainly wasn't when I was 25 - but only now. Strange, but true.