It is odd, it is a dilemma, it is frustrating. The Lyric has had official access since September 18 and we are both Ultium based cars. I have the adapter I linked above. It works. Setup an account on the Tesla app, including a credit card. Tell the app you are a Lyric with a NACS adapter. Initiate charging from the app, and when the app tells you to plug into the car, but not before. It works (on V3 chargers only. V2 is Tesla-only, and V4 have magic adapters and work on anything). It's not recommended nor is it official. It worked for me (the one time I tried it). The Prologue forums have found this out as well.
But even the SuperCharger network can be confusing. V2 chargers will only work with Teslas and we have a lot of them around me. So even with V3 and V4 access I find I still don't have quite the ability to do long distances that I thought I might have.
If only the industry had settled on one common plug from the very beginning. This was an avoidable problem.