Does everyone live nearby one another? Do the two year olds see each other a lot and play together? If so, perhaps the the 20-year-old mom is concerned that her child will feel left out. I don't know that this could be a concern at two, though it would be one if the kids were four years old or older. It would be you having to explain to the child why he or she is not going. This is even more difficult to explain if you are going to Disney or the San Diego Zoo. What can you tell a five year old? Your mommy has to pay for you? All the five year old would know is that he or she is not going. But since the kids are two, and if they do not see each other often, maybe this is not as big of an issue.
On the other hand, if the second mom isn't going, then it will be you having to change diapers, ensure naps, referee between the two kids, lug a kid through an airport or find room in the van for two car seats, That's not a vacation for the person in charge of the child. (It's already sounding less and less like a vacation, anyway).
In one sense, the 20-year-old's child could be seen as being left out if the rule is, "All kids under 18 get a vacation paid by Grandma," for you did say adults have to pay their own way. Or this can be viewed as, "All adults have to pay for their own children, but teens with children do not." It is interesting because the "rule" can be viewed as true from both angles, thus one can understand the 20 year old's disappointment.
Since the wording would have worked best if you had said, "Adults have to pay their own way for themselves and their families," you have your dilemma.
Point out to the 20 year old that you neglected to have the right wording, and that you never had expected to be responsible to be the main caretaker of any grandchild on the trip, and that you cannot do so. Apologize for the messiness of the situation and her hurt feelings, and that you never intended to hurt anyone's feelings.
If possible, ask her if you pay for the grandchild, is she able to pay for herself and then be the main caretaker of that grandchild on the trip? I know, you will end up doing a lot of the caretaking, but maybe if you create shifts with you and your husband going off by yourselves, you can achieve some down time.
Perhaps money is a concern for you, and you cannot fund the way for the second grandchild. (I do hear of grandparents funding the entire expense for vacations for all family members, but that's not my family, either).
Perhaps you are trying to establish the "Adults have responsibilities" rule, and the parameters of the paying falls under that idea. Perhaps you are trying to get them to be fiscally responsible and to realize that they have to save their own money for big ticket items because this has been an issue in the past.
What were the reasons for the parameters you put into place? Financial? Then I assume you have to say no because you cannot afford it. Getting the kids to be more responsible? Then perhaps pay for the child if the mother comes and pays for herself, and say no if she does not plan to go. Worries of jealousy because of who goes for free? Well, you've already lost that one.
Life isn't fair, and anyone who has become an adult knows not to complain about that when someone goes on vacation but he or she has to stay at home. The twenty year old is either talking (from her soul) like a child who hasn't learned that lesson, or she is saying it because adult children can at times revert to acting like children. It hurts to hear her say it because you were trying to be fair. I hope some of this post gives you the wording to help explain to her the ways in which you thought that this was fair (though she may not ever see it your way). Explain that it is not fair to you if you have to take care of her kid on a vacation. (Or suggest to leave the other two year old with her so you all can enjoy your vacation-ha ha!)
If you can afford to do it, and if you and the others would enjoy the kid's company, and if you can find a way to establish shifts for caretaking, take the kid.
That's a lot of ifs.