Don't get too worked up about it? It's 4 days; I know it feels like a lifetime (BTDT!), but it's not. Some kids are very, very determined and set in their ways (ask me how I know). With these kids, I always ask myself: what's the worst that can happen? At 18 months, she has other sources of nourishment, so she's not going to starve if she goes a week or two without milk (and you can probably sneak some in when you're making her food anyway if you're worried). So this isn't a battle you need to fight. Just put the cup of milk in front of her at mealtimes and pay no attention at all to whether she drinks it or not. I guarantee she can tell that this is a big deal to you, just based on all of the effort you are putting in to finding the "right" cup and the "right" way to present it to her. And she is rapidly reaching the age where her brain development is starting to tell her to reject what you push, because she needs to separate herself from you. So the more you focus on this, the more you ensure a continued battle. This isn't the fight you need to win, so stop worrying so much about figuring out how to "get" her to do this, stop making it such a big thing, and just start treating it as a normal part of mealtime.
(if you're worried about her getting dehydrated, (a) she's probably getting enough water from the foods she eats, and (b) you can always give her a cup of water between meals, or make lunch water and dinner milk, or whatever).