"called me for a meeting" and "coincide" are too formal unless the context is business.
In a business context, that is the correct formality.
For less formal, I would recommend:
I can't see you today because I have a family obligation at the same time we were supposed to meet.
I can't meet with you today because I've gotta go see my family.
Sorry, I accidentally double-booked (my calendar/agenda).
Also, 'arranged' is good but I would suggest 'scheduled'.