S.H.
... my kids, we taught them to go by their hunger cues and feelings of fullness. We do not teach them to eat for emotional reasons or boredom. We also do not 'make' them eat. If not hungry.
That is healthy eating. As you mentioned in your post.
Per your MIL... you need to ALSO teach your daughter.... how to say no, how to express her eating needs to your MIL. As a whole, with our kids, we taught them to speak up to our "overbearing" relatives. My Mom also lives with us... she grew up, always feeding people too.... as did her Mom and her Grandma. It is them. So... we TELL her.... not to overfeed and ALSO though, we do not have junk in the house. If our kids are hungry and if they want to eat a snack... its fine. AND at growth-spurts... my kids EAT a ton and very frequently, which we let them. And they grow like weeds and are very Lean/solid/tall kids. The Pediatrician said, if hungry, feed a child... especially when hitting growth-spurts. But to go by their cues for hunger or fullness.... according to their body.
We also let my Mom feed the kids though... she KNOWS by now.... what is fine. AND she eats healthy too. So, its fine. She does not offer them "junk." But, she does not 'nag' them about eating every 5 minutes.
Some people, show their affection, by feeding a child. Constantly. Which as you know, is not.... desired. My Friend's Grandpa, is like that. But he feeds her kids junk.
So for your MIL... maybe teach her to feed only healthy foods. Or make a list... that she can use.
My kids, if they are hungry, I do give them snacks. I do not have a 'schedule' for it. Per say. My kids KNOW their tummy.... and when/if they are truly hungry. They do not eat mindlessly. And they do not spoil their appetite for regular meals.
Remember... a "snack" can be even an apple. Or an orange or banana.
It is ALSO the food "choices" that are about healthy eating. It is NOT necessarily about the eating 'frequency'... that is healthy or not.
In fact... eating by "grazing" is VERY healthy, for kids and adults. This means, eating little portions throughout the day. Even if every 2 hours.
Now, because your daughter is throwing-up... due to OVER-feeding by your MIL... well that is just TOTALLY not good... if your MIL then keeps doing that to her. That is.... really awful.
Teach your daughter... how to speak up. To her. And how to say no.
If your MIL cannot do what is best for your daughter, and she is throwing up... this is real dysfunctional.
OR, GET a letter from your Pediatrician... specifying that your daughter is being harmed, by being overfed. I would, try speak to your Ped, and see what he/she says... for ideas on how to stop this.
all the best,
Susan