I reported earlier about my success in getting my UPS (re)detected by the NAS (FYI, I never changed/moved the USB cable in any way). But of course, something else has changed. The day after my update, I had a house power failure - and yes the UPS took the NAS down. And with power restored, it brought it back up as it has done for years. BUT this time, it didn't power on the attached DX510 unit - I had to press its own power button. I've always assumed that the dedicated eSATA cable provided the signalling to bring up the external cabinet - but not this time. This could be a one-off and I'm in no hurry to test it, but it's a bit of a co-incidence.
As to the other points regarding UPS's - my collection has grown to four, all protecting other non-computer devices therefore without auto-shutdown protection. These are non-brand units with very old bespoke software (unused/lost!), so I bought the APC Back CS350 just for the NAS because it was specfically supported by Synology. It currently reports 150 second power up time - I'm sure it was longer - perhaps time to change batteries?
I've had to replace ALL the sealed Lead/Acid batteries in ALL my UPS's over the years - never buy proprietary APC - ludicrous pricing! Two things: if you don't know how to deal with (potentially {in joke?}) high residual voltages, don't open the case! Also, the 'sealed' battery moniker is not always justifyed. I've seen the acid gel leak out - I was lucky to have replaced the batteries when I did otherwise there would have been a right corrosive mess leaking out of the UPS.