Interesting discussion! I couldnt help but pitch in my own two cents on this :P
First and foremost: do you know there are three types of teej fasts - incl hariyali, kajli, hartalika. While hariyali teej is popular in some states (incl Rajasthan), it is not so in others. Customs and rituals to be followed in each of these fasts differ - from region to region, community to community and family to family. What we are discussing is the teej fast kept in Anandi's family, in Jaitser village near Jodhpur district of Rajasthan!
Secondly, as opposed to Karwa chauth, the presence of husband is not mandatory - as in fact many women actually return to their parental homes to observe the fasting and festivities. While married women tend to fast for the long and healthy life of their husbands, unmarried girls fast to get a good husband (So it cannot be said that Anandi kept the fast only for Shiv or Jagya- theoretically. That Dadisa asked her to keep the fast for Shiv is a different matter)
About drinking water, since the norm is that once a woman initiates a fast, she should continue to observe it the same way as she did for the first time ; in many families, girls are advised to drink a glass of water so as not to be totally nirjala forthe 24 hrs for which they fast- as it may become difficult for them to observe nirjala fast during their pregnancy and/or ill health. (So Anandi may have done nothing wrong in drinking water). Or as some one else also noted- it may be a custom in rajasthan.
About keeping the same fast as the MIL - again - there are no hard and fast rules. My grandmother told my mother not to keep fasts if she didnt feel like, even though she her self kept teej. Eventually, my mother started Karwa chauth. The only thing to remember is that the practice of keeping a fast shouldnt be given up once started, unless one becomes a widow. So since Anandi had started observing teej (prevalent in Rajasthan), and since Shiv is also from Udaipur; Dadisa may not have thought of checking Shiv's family's customs before asking Anandi to keep the fast this year too!
Sorry for the long boring post...but as i said, couldnt help !! :)