1996 FIAT Punto starting issue

Hello
I have a 1996 FIAT Punto 75 with around 60000 Km, petrol engine 1.2L 75HP, that I bought used about a couple of years ago. I know this car is probably impossible to be found in USA, but engine issues are always the same :D.

The problem is that I think it doesn’t start very well. I said “I think”, because it has always been like that since I bought it, but I hear it doesn’t start as good as it should. Basically, what happens is that:
– if I start the car by turning on the dashboard one time, it doesn’t start well
– if I start the car by turning on the dashboard two consecutive times, it always starts perfectly

I did a video some time ago to show it, I put the english subtitles for you, here it is:

As you can notice, in the last case it starts much better than the first one. In fact, I always start the engine by turning on the dashboard two times :D. In the first case, it seems that the starter motor is like… I don’t know how to say it… “stressed”, it doesn’t sound good to me, and the engine increases RPM slowly. In the second case, the engine starts immediately without hesitation. Maybe this is normal and it’s just an obsession of mine, but I feel there is something wrong with it. I watched some videos of other Punto 75, and they seem to always start like the second case.

The starter motor is basically new, it has been replaced together with the battery around 6 months ago. Anyway, also the old starter motor did exactly the same thing.
Spark plugs, ignition cables and coils have been also replaced recently.

Some mechanics said it is normal and there is nothing wrong with it, others said it was the fuel pump. After having thought about it, I decided to replace it, especially because it was still the original fuel pump, almost 18 years old. However, after having replaced the fuel pump nothing changed again.

What do you think?
Am I crazy or is there really something wrong? 😀