Does the HP MPP 2.0 Tilt Pen really scratch the screen?

Hi,

I just bought the HP Spectre x360 16-f1002ns (i7 12th gen), and my computer didn't come with the pen in the box, so I had to buy one separately. At the store I bought the computer they only had the original 2017 HP Tilt Pen, but I returned it because the activation force was so high my handwriting was very choppy. I then exchanged it at the store for the Wacom Bamboo Ink (2nd gen), which is a lot better (even though not as good as it is on AES mode on another laptop I tried it on), and ordered the HP MPP 2.0 Tilt Pen to try it out and decide which one I prefer.

I have however now seen here in Reddit several reports of people who say the "included pen" (which I think is the MPP 2.0 Tilt Pen) scratches the laptop's screen, which have worried me very much. I was originally more inclined towards the HP pen (if it's at least equally as good) because I can attach it to the laptop's side, but I really don't want the only good laptop I've ever bought to be damaged.

Are screen protectors worth it or do they make the screen quality worse? Does the HP pen really scratch the screen or were they isolated issues? Does it also happen with other pens? What's the best pen for this laptop (preferably under 80€, although if I have to I'd rather pay a bit more than have a bad stylus), if it's not the one HP recommends?

Thank you for your help, I never thought choosing a stylus to just take notes would be so hard, as I had an iPad and was satisfied with a 25€ Apple Pencil knock-off.

tl;dr: Does the HP pen really scratch the screen? What stylus should I buy for this laptop?

EDIT (19-oct): I ended up buying HP's pen and a mate screen protector made by Upscreen, just to be safe. Thanks to everyone that replied for your help.