I was _____________________________ you wanted me to invite him.

The phrase "under the impression that" is the most idiomatic and commonly used expression to convey a misunderstanding or an assumption about something. It implies that the speaker held a belief that was not necessarily confirmed or accurate.

In the context of the sentence, choosing "under the impression that" suggests the speaker had a specific expectation or understanding regarding the invitation, which was likely based on previous information, discussions, or hints given by the other person. This phrase captures that nuance effectively.

The other phrases could express some level of belief or thought but do not convey the same depth of misunderstanding or assumption. "Believing that" and "thinking that" suggest a more active and conscious decision-making process, which may not reflect the implied surprise or confusion about the expectation. "Of the impression that" is grammatically incorrect and not a standard English expression, further making it unsuitable. Thus, "under the impression that" stands out as the appropriate choice for accurately depicting the intended meaning.