Career Break to raise children?

You could use the term "personal projects". This could mean a number of things - you were working from home on a personal business you were hoping to develop, you were working on your home, etc. "Family Business" will lead them to believe, falsely, that you were involved somehow in the business world. I would stay away from "homemaker" - which is unisex, by the way. Had you put housewife or househusband, that would give away your gender.

I'm not really sure how "homemaker" identifies you as female! There are plenty of men these days that stay home with their children. Or you could use the term "stay-at-home parent".

FYI, if you think that this organisation is not going to consider applications from females, then bear in mind that if you're invited to an interview they'll clearly see what gender you are! So if you really think that they make hiring decisions based on an applicants gender and that they don't want a female applicant, then a) you'll just get rejected later on in the process and b) not sure why you'd want to work for an organisation like that. Please note that I am not saying that you shouldn't apply or that you won't get the job - I am simply saying that it is ludicrous to try to hide your gender on an application for these reasons.

I wouldn't put "family business" if you weren't actually running a business. Let's put it this way - you don't ever want to put anything on an application that they could come back later and accuse you of lying about, and I think this term reasonably implies that you were working for a proper company with a proper turnover etc, albeit one owned by your family.

EDIT: In response to your edit, I have five years experience as a recruiter and I now work in HR so I play a part in hiring for my organisation. I have never actually seen anybody discriminated against on the basis of gender. And don't say "welcome to the real world" to me - I work in the "real world", whereas a stay-at-home parent clearly doesn't.

I'm pretty sure that homemaker would identify you as a woman, I'm not sure that I would put that down. And Family Business does sort of imply that you were working. Personal projects is vague enough that it can mean any number of things so that might be a safe bet.

Also, I'm fairly certain there's enough room in this here 'real world' for people doing all types of things, whether it being staying home and raising a family or being at a 9 to 5.

What does it matter if you're female, they're not allowed to discriminate (unless discrimination laws in the US are different from the UK). I'd put homemaker - family business sounds like you're trying to hide something.