What is the difference between UNIQUE, UNIQUE KEY and CONSTRAINT 'name' UNIQUE?

They are all synonymous as evidenced by syntax documentation:

[CONSTRAINT [symbol]] UNIQUE [INDEX|KEY]
      [index_name] [index_type] (index_col_name,...)
      [index_option]

[] in this notation (Wirth's notation) denote optional elements

1 and 2 are identical - both create two unique indexes, one for each key. #3 only creates one unique index across both keys, so no combination of username and email can be duplicated, but for example, a username could be duplicated as long as a different email was used.

Sounds like you probably want either of the first two. UNIQUE and UNIQUE KEY are equivalent.