All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
It is obligatory (Waajib) upon every Muslim to follow the way (methodology) of our righteous predecessors like the Companions and their successors and other scholars and people of excellence in matters of Aqeedah (creed) and actions.
Allaah Says (what means): {And whoever opposes the messenger after guidance has become clear to him and follows other than the way of the believers –we will give him what he has taken and drive him into Hell, and evil it is as a destination.} [Quran 4:115]
So all the good is in following their way, and all the evil is in not treading their path. We think that both of the groups mentioned in the question are upon this way (the way of the righteous predecessors). If the reality is one, then it is not important if the names differ.
Our advice to our sister asking the question and others is to love each other as brothers and sisters for the Sake of Allaah and not let a door open for the devil to scatter our unity and make us busy in things other than what is best for our worldly life and our religion.
The brothers and sisters should make an effort and be enthusiastic in calling respectively their brothers and sisters according to Islamic jurisdictions.
May Allaah bless this effort so that it would get its fruit and blessings by making people follow this true methodology.
Allaah Knows best.