Does "happen to" mean "actually?"

I happened to meet her this morning.

I've learned 'happen to' means to do something by chance, so the sentence means I met her this morning by chance.

A: I never seem to do anything right around here. B: I happen to think you do a lot of things right.

It's one of the lines in a movie, and someone told me 'happen to' here in the movie means "actually." I've looked up a lot of dictionaries, but I've never seen the meaning of "actually" for 'happen to.' What does it mean in that sentence, and if it means "actually," then is it used that way commonly?

Solution 1:

The OED has a nice entry (under happen, v) for this usage, which as noted in the comments, can be a bit confusing.

intransitive. With infinitive: used with varying degrees of intensity to support or imply an assertion. Also with non-referential it as subject, and sometimes followed by a subordinate clause

To support or imply an assertion is more tidily expressed in your question: the word actually is a very good way to paraphrase this. So yes, you have two different uses of the phrase presented in your question, and you have the correct definitions for both.