When does a person have a right to adverse possession?

No, what you describe is NOT adverse possession: "adverse possession" is, by definition, "adverse" to--or without the permission of--the property owner. If you were taking care of someone and her farm  while living with her, that is not against her will--it was with her permission. Therefore, this was not adverse possession and it does not give you any rights to the property.
The fact that the others did not help is irrelevant when there is a will: a person has the right to leave her property to anyone she wants, even undeserving people.
The fact that you put money into the farm is also irrelevant: no one forced you to do that; you chose to do it; and your voluntary choice to spend you money in this way does not let you claim rights over another's property.
If other people inherit and you do not, the ones who inherit have the right to remove you from the property: you have no right to say in another's property if they don't want you.
However, if you get a 1/6th interest in the house (1/2 of the 1/3 you and your sister share) you will be an owner of the house after the current owner dies and you inherit. That means you cannot be personally evicted or removed: any owner has a right to access, possess, and occupy the home. It does mean, however, that the other owners can force the sale of the home if they want, after which you will get 1/6th of the proceeds: the law does not require owners to retain a home they do not want but lets them compel a sale and distribute the money from it. So you do not need to worry about being personally forced out, but you do need to face the possibility of   the home being sold sometime after the current owner passes.
The current owner could give you more protection if she wanted. She could change her will to leave you the home entirely, because you have been caring for it; to leave you a larger share of the home and farm, so at least you'll profit more if/when it is sold; or to give you a "life estate" in it: the right to live in it (as long as you keep it up) for the rest of your life or until you voluntarily move out (such as to assisted living or a retirement community), at which point it goes to those she designated to inherit (which can still include you). So the only way to get more protection is if the owner chooses to give it to you.