Been a while since I played with my HT813 but I think the ring patterns are more for the FXS port ("how the phone should ring") than the FXO port ("the pattern incoming calls ring witih")
I did something similar to you, getting a hardwired condo enterphone to ring through to my PBX then through to my cell. I think it was a call forwarding setting on the FXO port and the call went through straight away. There's maybe a setting for caller ID on the FXO port too - this is usually sent between the first and second bursts of ringing, so you might want to turn that off to get a quicker notification of an inbound call.
Maybe post a screenshot of all the FXO settings?
Hmm.. lower voltage isn't the worst theory out there.. but at the same time I'd be surprised.. those ATA devices have always had super wide tolerances on the FXO port. What you could do maybe is spin up a syslog server on your PC and get the ATA to log to it for a bit. Go buzz in from downstairs and see if it spits out data on the ringing.
Your theory on the Xlink re-generating the ring tone? Yeah, could be - I'm not familiar with those devices but if you're going wall outlet --> xlink --> FXO port on the ATA and it's working? welllll... kind of suggests that's what's going on.. but I'm honestly surprised that wall --> FXO doesn't work the same way.
With it connected wall --> FXO, when you buzz in from downstairs, does it just ring and ring and ring? Or do you hear something else?