Did anything change when "I firmly resolve, with the help of your grace: 'to confess my sins, do penance and amend my life' (apparently with V2) changed to 'to sin no more and avoid the near occasion of sin'?"
I didn't realize this changed. I came up way after V2 and the Act of Contrition I learned was the one that ended with "I firmly resolve..."
Is it a lie if you feel your relations are a near occasion, but, if you're weak and/or demoralized by living in a licentious atmosphere, moving to a new town or joining a religious community is not going to work out at some time or another. Is it like a lie, then?
I don't think its a lie. Many a saint and holy person lived around people and areas that were obvious "near occasions", the difference was they were able to resist the temptation in their own persons and reach out to try to change the world around them. If it is impossible for you to remove the obstacles, you have to ask the Lord for the strength and virtue to resist them. And if you stumble, that's why we have confession.
I never really got the "firmly resolving to confess my sins part" as you're already confessing your sins at the time, so you probably believe in it or your parents or private school is making you do it and they probably will again if you don't go regularly.
I might be wrong, but I always took it as 1) if you are being made to go by parents or school, to resolve to do it on your own, 2) it is a confirmation of your beliefs in the Catholic Church since Protestants churches for the most part don't even believe in private confession and 3) You can and should say the Act of the Contrition outside of Confession, too, and that would be a resolution to go.