Krótko mówiąc:
Take out to neutralne wyjmowanie czegoś (np. z torby, lodówki). Jest to najczęstszy i najbardziej uniwersalny zwrot.
Pull out sugeruje użycie siły lub pokonanie oporu, aby coś wyciągnąć (np. ząb, samochód z błota, coś co się zaklinowało).
Prosta zasada: Jeśli nie jesteś pewien, użyj take out.
Hello. I've heard native speakers use both phrases (take out and pull out) in the same contexts, for example I took out my phone and I pulled out my phone, so in terms of communication, I would consider them more or less the same: there's no need to overcomplicate things. Context is what matters most anyway – dictionary definitions don't always work in real life.
The verb "pull" can also mean something like "show", as in "Could you pull up the video real quick?", meaning "Could you show the video real real quick?".
Hope this helps :)
Yes, for taking a book or phone from a drawer, "take out" and "pull out" mean the same thing and you can use them interchangeably.
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"Took out" = more general/neutral.
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"Pulled out" = very slight sense of physical motion.
Both are correct for wyjąć/wyciągnąć in this case.