Native English speakers: What is the meaning of "wrote a reference letter for me"?

What is the difference in meaning between these two? (a) Mr. Jones wrote a reference letter to me. (b) Mr. Jones wrote a reference letter for me. My understanding is that (a) means the same as Mr. Jones wrote me a reference letter." Which means that Mr. Jones wrote a reference letter and I got it. (B) means... show more What is the difference in meaning between these two?

(a) Mr. Jones wrote a reference letter to me.
(b) Mr. Jones wrote a reference letter for me.

My understanding is that (a) means the same as Mr. Jones wrote me a reference letter." Which means that Mr. Jones wrote a reference letter and I got it.

(B) means that I was originally asked to write a reference letter to someone but since I was busy, Mr. Jones did the job instead of me. Or, (b) could mean the same as (a).

Is my understanding correct?

Thank you. I'd appreciate your feedback.
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