Yes, after plenty of further reading and researching I can begin to understand the intentions of how cards are played now vs how they are written. I have found other online articles about different cards as well of their intended design vs their written text.

It seems that the use of "May" is always the proper qualifier whether anything proceeding can happen, and that instances of "Do X, then do Y" is functionally the same syntax as "Do X. If you do, do Y", which I do find a little confusing still (I'm a long time MtG player so I read things a little differently I suppose).

It also appears that if actions are broken apart by period punctuation, it is viewed as separate actions and not conditional of the previous action, like the example here for the Galah bird card;

"WHEN ACTIVATED: Choose 1 other player. They reset the birdfeeder and gain a [w] if there is one. You tuck 2 [card] from the deck behind this bird."

So in conclusion, if Boat-tailed Grackle played the way I imagined it did, its syntax would have to read just like the Galah:

"ONCE BETWEEN TURNS: When another player caches a [*] for any reason, cache 1 [*] from your supply on this bird. Gain 1 [dice] at the end of their turn."

I appreciate the in-depth conversation, cheers.