No, first party means it's developed in-house (Super Mario, Mario Kart, Zelda, Metroid, Animal Crossing - all developed by Nintendo EAD, Nintendo R&D1, or another studio with "Nintendo" in the name)
Second party means it's developed by another studio, but the IP is owned by Nintendo (Fire Emblem, Xenoblade, Kirby, Pokémon. All developed by their respective companies I'm sure you're familiar with). A second party game actually doesn't require you own the studio, but yes, Nintendo do own some of the studios of their second party games.
Third party is just completely unaffiliated in any official capacity and their games are licensed to be on the system (SEGA, Capcom, Namco, etc)