It has been more than an hour since the opening post... 
AMEX only cares that you pay them. If you don't pay them, they get kind of snippy for some reason. But as long as they keep getting money to pay your balance, they are happy.
BT of this size are not going to be looked at by anyone, they are small. If you were trying to regularly BT $50k around, say 4 or 10 times, and your typical balance was $1k, you would get a phone call most likely. For these transfers, no one cares enough to call 
The AMEX FR is more likely to turn up when you are bringing debt onto the AMEX card. Paying down, or BT off to another card, seems that's on the "good side" of their equation.
And yes, I've done a few BT off of my Delta Gold card recently to wind down the low APR period, and they just show up as "electronic payment", which is what the BT Destination bank is doing. As far as I know, there is no transaction designator in the ACH payments system that does much to identify the type of payment involved, probably not able to identify a BT different from a regular payment.
That's probably an important point to make. ACH = Automated Clearing House. The transactions are not directly from one bank to another, they have to go through the central payment clearing hub, so that creates certain requirements to make transactions generic.