"Taking responsibility for your actions" is an admission that the subject and nobody else is at fault, that's correct. In particular it's an acceptance that the person concerned expects to be punished in some way. Whether they are punished or not, and whether or not they ask for mercy, are actually separate matters from this point of view.
Note however that this is not the same as taking responsibility for fixing the consequences of your actions. It may be the case that someone other than the responsible person might be better at fixing things up, or the person responsible might no longer be in a position to do anything about it (in a different job, perhaps), or the fact that it was that person's fault may not have come to light until after somebody else fixed the problems.
However, this is only one use of the word responsible, case 4 in wiktionary's definition. It sounds like you were originally assuming something closer to the sense of case 3: roughly "being in charge of" something. In this case nothing has happened yet that can be the person's fault!