[@mention:560742421255289227], great questions! We find that the name of the role/circle should convey identity and intention while the purpose, if applicable (as per [@mention:561164855662986246]'s insight), should inspire.
For example, your name appears to come from the Greek "Eugenes" (which means "nobility, well-born"). While that should inform your purpose, it may not be as inspiring as a well-crafted purpose statement, which should steal your breath a little (in a good way).
We've seen that purpose limits output, which is why it's important to have an aspirational purpose statement — one that really matters if not achieved and has the potential to move the needle (think "worthwhile and meaningful for those who expect to change the world"). It's hard to write a great purpose statement. It's impossible to pursue a great purpose without a concrete and shared understanding of the purpose.
Concerning two circles with identical purpose, I would want to know a little more about the work of the circles before offering any ideas.