Look I'm frustrated with the internet community at large. And honestly with how I absolutely think fit and non fit and former fat or even still fat people are being judgmental self righteous assholes about fat people MUCH too often. I get that you have your own frustrations, but what you're saying doesn't really go against any grain at all except for the subsection of people you live with. I reacted negatively because I always see people use this as an excuse to be assholes to fat people. I'm honestly shocked by the number of people that just want to be mean to fat people because they're fat.

Your families reactions screams to me an example of the larger problems at play. They use this woman as a way to feel better about themselves because they are not where they are at mentally at all with regards to their bodies. They are truly insecure about it and react defensively.

Here's how I think they see it. They see people saying come to the gym with me not as "come to the gym with me" but "you're broken and unlovable and need fixing, let me do it." My mom used to make comments about my weight all the time. She still does. So whenever she would suggest hey come on a walk with me, what I heard wasn't "Come on a walk with me" I heard "You're fat, let me fix you so I can get the daughter I always wanted." I refused every time because I didn't want someone in control of my life. I got tired of the idea that I was broken. I was in the normal range, but overweight now. I'm losing weight pretty rapidly at this stage in my life and while I love my mother she has no idea about it at all because I just don't want to hear her well intention-ed advice about fitness.

Don't "coax" (sorry that just brings up all the things my mother use to do) them, just put the idea out there. Hey you're welcome to come to the gym with me anytime, just let me know. If it's such a point of contention, tell them "I don't appreciate when you say I want to look like a super model. That's not my goal just insert whatever goal you have is", let them know that they are hurting you when they say that. The people in your life are going to figure this out at a rate which is different from yours. Doesn't make you better or them worse, it's just how it is. You're well intention-ed like my mother, doesn't always mean people see you're good intentions. They have a lot of insecurities and shit to deal with. Some people won't, at some point you will probably have to let it go because it does you no good to try and keep convincing them.