I am FINALLY able to get essentially the behavior I wanted, but I'm getting it by virtue of Google Voice and not the Obihai device. Someone produced a Firefox extension similar to Telify but specific to Google Voice; it uses a Google Voice account and the callback feature to initiate a call and ring a specified phone number. Since it says I can't use the Google Voice number itself (at least not one for the same account?), I used my landline number which arrives at the Obi110 at the LINE port.
Now when I click on a phone number in a Web page, it dials the number via my Google Voice account and rings my landline, which rings the same phone attached to the Obi110 that I had wanted to ring via the Obi110 itself. Since I now have two other GV accounts with numbers, I might be able to use one of the other accounts to configure the extension and then specify my primary GV phone number, the one associated with SP1 on the device, and have it ring the phone through the GV number instead.
It's not as elegant a solution as one that communicates directly with the Obi110 via the local network to initiate a call, but at least it works. Since discovering that someone had written a WSH script to initiate recording an active call, I had thought that what I wanted should also be possible, but I'm disappointed to find that it's not implemented even if it's possible.