I would use "avgjøre", and in some cases maybe "bestemme". Compare these sentences, for example:
1) Utdanningen din påvirker hvilket parti du stemmer på.
2) Utdanningen din avgjør hvilket parti du stemmer på.
In 1), a certain (level of) education increases (or decreases) the likelihood for voting for specific parties. But the effect is not deterministic, so there has to be other factors that influence voting. In 2), party choice is completely determined by education.
That is at least how a political scientist would use the terms. Journalists may use 2) to express the meaning in 1), to make their headline more catchy.
Ah ok, I see!
The words in question, however, differ form each other along more substantial semantic and syntactic dimensions apart from a trivial question of implied presence or absence of external factors. They can't be used interchangeably.
a) Utdanningen din påvirker deg.
b) Utdanningen din avgjør deg. (nonsensical)
c) Han avgjorde at de skulle bo der.
d) Han påvirket at de skulle bo der. (nonsensical)
Also, when you say that
påvirke implies other factors (that is, it is an inherent part of the meaning of the word), then the following sentence e) should be bad in your ears while f) is good. Note that both exclude other factors by the presence of
bare. My guess is that you accept both but that they have different meanings.
e) Bare han kan påvirke utfallet.
f) Bare han kan avgjøre utfallet.
Lastly, I have no problems using
avgjøre in contexts where it is contingent on other factors.
Hvis du er kvinne og tjener mer enn 500.000, avgjør utdanningen din hvilket parti du stemmer på. This implies several factors and could easily be expanded, thus party choice is not solely determined by education. The reason I think you get the impression that
påvirke implies other factors whereas
avgjøre does not, has to do with the discrete outcome of
avgjøre which is not shared by
påvirke which operates on a more gradual scale.