If we do loans to directors or officers of other institutions, are we required to report these loans in some way?

No.

Okay....we have an option in our core software to designate individuals as "Officer of other bank" or "Officer of other affiliate." If there is no particular regulatory reporting requirement for reporting loans to directors/officers of other banks, I suppose this is simply for our own internal use?

You might want to ask your software provider; but it does appear this is so you can keep track of loans to officers of affiliates (banks and other companies).

During a Bank exam the examiners will usually ask for a list of loans made to officers and directors of other banks.

The regulators would be checking to make sure that you are not granting favorable terms to any Directors or EO of correspondent banks. The ability to identify these loans is not a bad idea.

Here is a Fed comment regarding this issue, from the Federal Register in 2007. The comment is located here

"The Board also has received numerous inquiries about how a bank can ensure compliance with the correspondent lending restrictions in 12 U.S.C. 1972(2), given that all related reporting requirements are being eliminated as part of this rulemaking. Briefly, the correspondent lending restrictions in 12 U.S.C. 1972(2) require, among other things, that extensions of credit by a bank to an insider of a correspondent bank be on market terms. In light of the elimination of the statutory and regulatory reporting requirements associated with 12 U.S.C. 1972(2), a bank may select any reasonably prudent method to ensure compliance with the restrictions. For example, a bank may establish policies and procedures to request additional information about a borrower's relationships with correspondent banks when the bank determines that a prospective extension of credit to the borrower will be on preferential terms."

Is there any special language that must be in the Note for loans given to officers or directors of other banks (that are not a correspondent or affiliate bank)?

No. There is no specific language that has to be in the notes, even if they are correspondents. The only specific langauge in notes that is required by Regulation O is a demand clause for the those individuals that are deemend the bank's own executive officers.