They also have a service where you can watch any movie on Tuesdays for $5, for $12 a year. I prefer that service, and have renewed for 3 years.

Does this service have a name?

AMC Stubs https://www.amctheatres.com/amcstubs

I've been a subscriber for a few years. They waive online ticketing fees and you get $5 vouchers for every $100 spent (among other perks). It's been worth it for me.

The base model of stubs is free. You still get $5 Tuesdays.