According to dictionary.com:
VILLAIN
–noun
1.
a cruelly malicious person who is involved in or devoted to wickedness or crime; scoundrel.
2.
a character in a play, novel, or the like, who constitutes an important evil agency in the plot.
According to both definitions, both Sweeney and Mrs. Lovett are villains, as are Judge Turpin AND the Beadle. Note, though, there is nothing to say that the villain can't be a protagonist, which Sweeney clearly is.
I would argue, however, the witch in Into the Woods DOES NOT meet these definitions for villainy. She is NOT cruelly malicious and devoted to wickedness. Whatever malice she does (locking Rapunzel in the tower, banishing the prince) are not because of her devotion to wickedness (as Sweeney and Lovett's pies are) but rather motivated by much more specific goals.
Even her spell when she hurts people isn't really wicked in intent because it's so minor.
According to the definition, the only character in Into the Woods I can even see as a villain would be Cinderella's step-mother and sisters...
Hi, Shirley Temple Pudding.