For example, if I were in a situation where I said something and it was misunderstood (maybe because of bad Japanese), what would be the best way to attempt to correct myself?
In English one could say, "Sorry, I meant ..."
e.g., Saying, 「これは兄がもらった贈り物だ」(This is the present my brother got) when you meant, 「これは兄にもらった贈り物だ」(This is the present I got from my brother), and you want to tell them this correction after the other party has replied.
What would be the equivalent phrase to use when correcting somebody else?
In English it could be, "Did you mean ... ?" or, "I think you mean ..."
e.g., Someone says to you, 「これは兄が私に上げた贈り物だ」 and you want to correct them to say, 「これは兄がくれた贈り物だ」 (presuming they didn't know they made this mistake)