Nobody has mentioned that you need to buy the variety of potatoes for frying or baking.

In German you should buy 'Mehlig kochend' potatoes, in English they are called floury or mealy.

https://de.wikipedia.org/wiki/Liste_von_Kartoffelsorten European popular potato varieties

The usual alternatives are called waxy, (in German "festkochend") and do not achieve the very soft insides,
and so they are used for potato salads.

Floury, or mealy (baking) potatoes have more starch (20–22%) than waxy (boiling) potatoes (16–18%).
The distinction may also arise from variation in the comparative ratio of two potato starch compounds.