Found your post searching for answers to GEN 2 MYSTERY GIFT. I've been doing mystery gift for just under a year now, wondering if I could get every gift on GOLD using STADIUM 2 alone as I've only got 1 GameBoy Colour.
I compiled and ran the mystery gift tool you referenced using my GOLD trainer ID (17817).
I have been getting several gifts which do not align with those my trainer ID can send out. For the first few months I received MUSIC MAIL very often, despite that being a "no" using that tool. I've also received GOLD BERRY, and several Dolls that my ID does not send to other GBC games according to that tool. Of the 11 Dolls I've received thus far, 6 are possible on my ID according to the tool, 5 are not.
I can only assume the gifts from Stadium 2 are not limited to trainer ID in the same way, though there may be some correlation in probability?
Some anecdotal thoughts to add. Only within the past 3 weeks or so I have a CRYSTAL cartridge which I've started playing. So far that game has received 4 dolls (1 of which GOLD already had). It may be that when one of the games gets a rare gift, the other cartridge tends to get a better item too. I've not nearly enough evidence to back that up, but figured I'd mention my feelings so far based on your comments.
Finally, the STADIUM 2 character does mention the TENTACOOL DOLL very very often! Thus far I've not received any of the STADIUM 2 exclusive items on either GOLD or CRYSTAL cartridges (PIKACHU BED, UNOWN DOLL, TENTACOOL DOLL). I've no idea if any special triggers are involved with those....
I'd love to know if you have since found any documentation on how this all works?