I found one article in the Infovault that says wire transfers count towards the savings transaction limitations and another article saying that it was dependant upon how the request was originated. (via phone, in person, mail, etc.) Which one is it? The next question is---do we consider it a check or preauthorized transfer?

A wire transfer is not a transaction on an account. It is a method used to transfer funds that are withdrawn from the account. Since I think I wrote the "second opinion" you cite, I'll stick with that opinion.

What you need to focus on is how the transaction was effected.

To segue on this question. We have a customer (County Board of Commissioners) who send a file of ACH transactions in weekly. This is debited from their money market in one lump sump; however they also have 4 other transactions for the month. The ACH file comes into the Bank through our Internet Banking Department.

OK - do these count?

Yes, the file counts. As one transaction, because there is only one charge to the customer's account.

I liken this to the Fed Staff's discussion about how to count transfers from a savings account to cover a DDA overdraft when there are multiple checks involved in the overdraft. The staff said that the answer depends on how the transfers from the savings account are made. If one transfer is generated at end of processing to cover the aggregate overdraft, it's counted as one transfer. If a transfer is created for each check that generates an overdraft, each transfer is counted separately.