I tried to search the internet but didn't find any information.
I'm not aware of any specialized term that set apart fountain pens and dip pens, both are called in Russian "перьевая ручка". However, when one wants to describe the exact mechanism of how ink is refilled, fountain pen is called "заправляемая перьевая ручка" and dip pen is called "незаправляемая перьевая ручка".
The pen that you have to dip in ink once in a while to continue writing is simply called "перо́" ("пе́рья" in the plural):
[...] пе́рья Пу́шкина отме́чены осо́бым зна́ком вне́шней похо́жести, одина́ковости – сло́вно всю жизнь поэ́т писа́л одни́м и тем же перо́м.
"Перо́" literally means 'feather' as goose feathers were traditionally used for writing. But a "перо́" does not have to be a feather, it can be made of wood, steel, etc.
Pens that have ink in them are called автору́чка or simply ру́чка. If you want to be more specific, use перьева́я ру́чка (fountain pen) or ша́риковая ру́чка (ballpoint pen).