EEA Family Permit for Student questions

Hello! Me( Greek national) and my civil partner(Ukrainian national) are going to apply for an EEA Family Permit. I’m a bit confused regarding the CSI and we’d highly appreciate it if you could help(we’re applying in Greece): We’re going there cause I’ll be doing my Master’s. We’re planning to send my acceptance letter as a supporting document. I know that students and their partners need to have a CSI. We both have EHIC that will expire in February, a month before the end of the 6 month period of the Family Permit.

Will I already be considered a “person exercising their treaty rights” or that could only be relevant after 3 months?I’m asking cause I know that if I was already there for 3 months, we would both need a CSI,but since we’re going to be travelling to the UK together and it’s basically impossible to get a CSI without being there, I’d like to know if we have to show a CSI at this stage or it’s not necessary
2)Would it be better to avoid mentioning my studies just to avoid confusion? If I understand correctly, there’s no need to say why what’s the purpose of our visit.

Sorry for the long post, we’re quite confused. Thanks a lot for your help in advance