First, the thread title is an instant inductee in my forum-thread Hall of Fame. Nice.
Your description of your 2BK (with the recovery movement after the propulsive flick) sounds just right. It sounds like you are bringing the feet back together and into streamline after the working beat -- great. What makes it two-, four- or six-beat is the number of propulsive downbeats per stroke cycle. If I understand your description, you are doing a 2BK.
On the hammy thing, you might try to get that recovering upsweep accomplished with hip muscle (glute, working in hip extension mode), just like you want to initiate the downbeat mostly with the opposing muscle group (hip flexors). As long as you are not doing a leg-curl type affair on the upbeat, you are good to go. Skating drills are a good occasion to pay attention to which muscles are driving the leg movement and feel the differences.