Both are correct for all the sentences with 'eisiau'.
There is some overlap - eisiau has a broader meaning than just English "want", but "want" and "need" aren't both correct for all sentences. It depends rather on the construction, for example:
Mae Siân eisiau canu would be "Siân wants to sing" not "Siân needs to sing"
Mae eisiau i ti fynd would be "You need to go" rather than "You want to go"
Mae eisiau car arnoch chi could be either "You want a car" or "You need a car"