Modern Vespa

Good Morning,
Recently I bought 2 second hand Vespa GTS's. A 2008 - 125cc and a 2012 - 300cc.
The 125 had slamming handlebars in the beginning, but that was solved by removing the top case and changing the handlebar weights.

However, I continue to feel that the 125 steers worse. If you turn a little, the handlebars "fall" into the corner, you actually have to hold back not to steer further. Something I'm not used to with motorcycles and the 300 doesn't have that either. If I change the wheels of both scooters, it's gone.
However, remarkably, the tire of the 125 is certainly not worn, because if it was driven "square" it would be explainable.
The 125 does have Michelin City grip 2, while the 300 has Michelin City grip Winter on it.
Is there much difference between these tires per se in terms of shape and handling? Does anyone have any experiences with this?

I think the tyre is mounted in the wrong direction on the red scooter.

Can the direction of the tyre have such a steering impact as well?

When I changed the front tire on my GTS 250 to another brand I noticed that the handlebars would fall into the corners as you described. The only other time I have seen this is on a rental scooter that had obviously run into something and had the forks bent back a small amount.

There is a sticker on the Vespa, ignored by everyone, telling us exactly which replacement tires to buy. Perhaps there is a reason for this.

Cheers and good luck,
Bob

As above, I bet the steering column of the 125 has been bent back a bit, causing noticeable oversteer.

Oh, and the front tyre is on the wrong way round. I doubt that affects the steering much though - but get it changed and report back!