How to correct myself or somebody else when speaking

Question

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)

Answer 1

For the second sort of grammar correction, I go with:

もしかしたら、「correct or what I imagine to be correct thing」でしょうか。

For the first type:

すみません、「what I meant to say」と言いたかったんです。

If what I said wound up being rude:

失礼いたしました。違うこと、「correct thing」、と言いたかったんです。

Answer 2

I often just use/hear じゃなくて〜.

これは兄がもらった贈り物だ。あっ、じゃなくて、兄にもらった贈り物。

Also, I know of [元]{もと}へ, but I don't know how much it's actually used. Here's an example my dictionary gives.

次の計算問題は22プラス12…元へマイナス12です　→　The next problem in calculation is 22 plus 12... no, [I take that back / correction] ─ (I mean) minus 12.

I also seem to remember hearing 改めて(言う)... from some people, but I might be making that up.