According to the understandings of new age thought:
All these relationships don’t have to be sexual, or romantic, and can be platonic.
Twin Flame: Is a another entity that is made from the separation of your soul, and is literally another piece of you. This understanding originally came from theosophy, and twin flame is closer to the original definition of soulmate.
Soul Mate: Are predetermined entities that you come in contact with that will teach, and guide you throughout your lives. You are able to have more than one soulmate. Soul Mates are considered to be part of your soul family. This understanding originally came from theosophy, and did not gain this definition until it was modified by the new age.
Karmic relationship: Is a relationship with someone else that is very intense that is not supposed to last. You are able to have multiple karmic relationships. They are bad relationships that are meant to teach you lessons about what not to look for, and it is thought to be the result of karma. This understanding originally came from new-age understandings.