In Italian, for family memebers and relatives: your mom and dad, grandpa and grandma... (singular), you have to omit the article il/la, however in the plural form, the Italian definite article will appear before the Italian possessive adjective that refers to a family member or relative.
Mia nonna è vecchia. (My grandmother is old.)
Le mie nonne sono vecchie. (My grandmothers are old.)
Note: Your fiancé/ finaceé is not counted as a family member yet, so you still have to use the article :)
If I may, i should say that the rule you explained is correct for all the members of the family except for Mom and Dad: in this case you can use or not use (indifferently) the definite article. For example you can say:" il mio papà è alto" or "mio papà è alto" both of them are absolutely correct, or, "la mia mamma è bella" and "mia mamma è bella".