House rules are defined as what rules are in play at a given location or house, suchs as a bingo hall, a casino or a poker game at someones house.
Variant rules is the term since at least the 3rd edition of Dungeons and Dragons for optional rules different from the core rules. Optional rules could work too, however, at the point the point they are an established house rule, they are no longer optional.
Homebrew literally means an alcoholic beverage brewed at your home, but generally means content you created. Although my homebrew is not your homebrew, collectively this content is homebrew.
Third Party means a creation by a publisher intended to be compatible with a product by another publisher.
Thus a variant rule or optional rule can come from the publisher (first party), be a homebrew creation by the game master (second party) or come from a third party (third party publisher or a homebrewed option from another source than the second party). Whether or not any given rule is in effect is governed by the house rules.
"Unearthed Arcana: Variant Rules" from 2015 by Mike Mearls:
For this month’s Unearthed Arcana, we’re showing off a few variant rules that you can use in your campaigns.
Up for discussion are the options of [...].
The material presented in Unearthed Arcana will range from mechanics that we expect one day to publish in a supplement to house rules from our home campaigns [...].