a modification agreement and a new note
If you are truly modifying the existing loan then why is there a new note? Only a modification agreement would be necessary. If the new note is now your legal agreement then it replaced and satisfied the existing note therefore making it a refinancing.
You must REPLACE an existing mortgage with another
I'm sure this is a mis-quote on TR's part. Satisfying and/or replacing the mortgage has no bearing on the definition of a refinancing. The obligation (note) must be satisfied and replaced. See the following FAQ at the FFIEC's Web site.
Refinancing --- satisfaction of lien. Is the satisfaction of a lien (mortgage) relevant to determining whether an obligation is a reportable refinancing?
Answer: No, the satisfaction of a lien is neither necessary nor sufficient to create a reportable refinancing. The credit obligation must be satisfied and replaced; it is not relevant whether the lien is satisfied and replaced. See 203.2(k)(2)