The FAQ section on departuresoffer information on the basics of EEA visas, as well as others.

EEA family permit
EEA Family Permit Requirements (for non-EU citizens outside the UK):
The EU citizen intends to exercise their Treaty rights in the UK (employment, self-employment, study, self-sufficient, etc)
The couple must be legally married to each other / in a civil partnership with each other
They must have met each other
They must intend to live together
The couple must be traveling to the UK together or the EU citizen should already be in the UK

EEA2
For non-EEA members, an EEA2 is a residence card that confirms their right of residence under European law as a family member of an EEA national. A family member is a spouse or partner, dependent children, dependent parents and grandparents.

I hope that helps clarify which visa you should apply for. Anyone have personal experience?

 

The part time employment shouldn't be a problem, as the rule stipulates that one should be in employment (and does not specify nor discriminate between part and full time employment). The underlying reason for this is the government intends to make sure the applicant/s will not be a burden on the government.