Should a paper of exactly 15 pages be submitted to Transactions or Proceedings of the AMS?
You should submit your paper to Proceedings.

The overview page for Proceedings of the AMS clarifies this:


  This journal is devoted to shorter research articles (not to exceed 15 printed pages) in all areas of pure and applied mathematics.


If your article is exactly 15 pages long (or less), then its length does not exceed 15 pages, and so it is suitable for submission to Proceedings.  By that same token, it is not suitable for submission to Transactions.  The intention is clearly that the criteria should be mutually exclusive.

Information on a journal's own web page should be considered more authoritative than information about that journal on a different journal's web page.