Elokeinu is the type of relationship Hashem has with us. He chooses to relate to the world as a king, i.e., Melech Ha’Olam, He chooses to relate to us as a father as well. In Chassidus, and this is a general principle, Elokeinu means “our strength and our vitality.” In Loshon HaKodesh it’s “kocheinu v’chayuseinu,” which connotes our physical life itself, our energy, enthusiasm, life dedication, etc., as explained at length throughout Chassidus.

For example: Reuven is a man. “Reuven,” is just his name, the “handle” to refer to him. To his wife, relates as “husband,” to his chidren, his relationship is, “father,” to his employees, “boss,” etc.

Havaye is G-d’s name, the “handle” to refer to Him. His relationship with us is “Elokeinu,” or “Avinu (father),” etc.