FWIW, I found a work-around for the problem of CID not displaying properly on my two-line phone. I use a device called a Ring Selector by Aastra to direct the calls to the appropriate line based on the ring pattern. Calls to my primary number (standard ring) are directed to line 1 on my phone, whereas calls to the virtual number (distinctive ring) are directed to line 2. When I do that, CID displays properly for both lines.

This setup means that a second call to either number goes to voicemail rather than to the other phone line. This loses some functionality, but we prefer it this way (we don't like or use call waiting either).