Does this category include all the level as the ones that you unlock after beating other on Glaring difficulty, the V Fields, the compass ones, etc... (I may forgot some) ?

For the "Beat the game" category, you do not need to complete all fields, only finish field X5, at the entrance to the spirit forge.

For the "complete all fields" category, you do need to complete all of the fields, which includes the vision fields, fields unlocked by completing certain fields on glaring difficulty, and compass fields. On the steam version, there are a total of 191 fields. This site contains a list of all fields: http://gemcraft.wikia.com/wiki/List_of_Fields_in_Gemcraft_Chapter_2.

Okay, thanks for the answer, that's what I thought and the "complete all fields" seems interesting but pretty long and I have some difficulties on some visions, and I still have to use compasses to unlock those levels. I'll try it later I think. And thanks for the link too :) !