UPDATE #2: I also got the achievement for not receiving damage in a entire episode (the speedrun + no damage achievement popped up at the same time after completing the game). The weird thing is that I took some damage in the last boss fight so yeah, it is clear that the achievements are bugged.

UPDATE: I am sure I was a little bit slower in my second speedrun because it wasn't my intention to go fast since I had 0 upgrades (needed to be a little bit more careful, you know) but I got the speedrun achievement anyway. Super weird. Another weird thing is that I did not get the achievement for completing the game without upgrades. What a buggy mess. I am starting to think that I got the speedrun achievement by pure luck