The cause is a setting in Audio mode: "Turn off background music", under the cog wheel next to the playback buttons.
When that setting is switched off, Remember Me registers as a secondary audio app on iOS so your podcast can keep playing. iOS gives the Lock Screen Now Playing card only to the app that owns playback, so as a secondary app we never get it. That accounts for all three things you saw: no card on the Lock Screen, the AirPod squeeze going to your podcast instead of to us, and the two audio streams playing over each other.
Switch "Turn off background music" on, then start playback from a box and lock the phone. The card appears with ⏮ previous verse, ⏸ pause / ▶ resume and ⏭ next verse. The trade-off is that other audio stops while verses play — that is what owning the Lock Screen controls costs.
Two changes are queued on our side:
- the setting will tell you that keeping background music means giving up the Lock Screen controls, instead of trading them away silently
- the play/pause button on AirPods and headphones is not wired up yet, even with the setting on. Squeezing an AirPod is the gesture you were after, so that is getting fixed.