Hmm... I always thought "at dinner" means "during dinner", so like "talking at dinner" etc. And "for dinner" means "the food which is our dinner". Just tried to search it and couldn't really find exactly this.
You really think "eating meat at dinner" should be added? I assume you are both native speakers right.
Mи sounds more like mih, but the ih is kind of deep sounding, I don't know how else to explain