Overall one can't deny the influence of expectations.
After reading more about the matter, I'd say that Pygmalion effect (or Rosenthal effect) only speaks of positive expectations others might have on us, so it doesn't apply to
- negative expectations.
- expectations we have for ourselves.
For the negative expectations there's the Golem effect and according to Wikipedia it is
a psychological phenomenon in which lower expectations placed upon individuals either by supervisors or the individual themselves lead to poorer performance by the individual.
In this case it's irrelevant who has the expectations (if they come from outside or from ourselves). Also, from Babad, E. Y.; Inbar, J.; Rosenthal, R. (1982), "Pygmalion, Galatea, and the Golem: Investigations of biased and unbiased teachers", the effect
represents the concerns of social scientists and educators, which are focused on the negative effects of self-fulfilling prophecies
Notice that the German-born American psychologist Robert Rosenthal is behind both of these effects.
Then, and according to Psychologinie, there's the Galatea effect which is
a phenomenon where people's own opinions about their ability and self-worth influence their performance.
The following image (also from the same website) pretty much describes the difference between Galatea and Pygmalion effects