I didn't post that picture, she did.
Which of the two sentences is correct?
Or are they both right grammatically?
Do they mean the same thing?
I had logged into the account using her phone, and [then afterwards] forgot to log out.
Those tense choices draw attention to the sequence of events, both of them in the past, one happening before the other.
You can also say it without corroboration from the choice of tense, allowing the listener to infer the sequence:
I logged into the account using her phone and [then afterwards] forgot to log out.
In situations where relying upon the listener to infer would be sub-optimal, use the past perfect, inasmuch as it makes the sequence explicit.
You don't want to use past perfect in both clauses since it contributes nothing to the meaning in that case, unless you were to tack on yet another clause, and wished to assign it to a different time-frame, grouping the first two clauses as having happened in the same time-frame.
You're talking about two completed actions (logging in, then forgetting to log out) both of which happened in the past, so I would use past simple in both cases as follows:
I logged in to my account using her phone and forgot to log out.
