Well after a bit of googling, the Oxford English Dictionary defines a home as a domicile / place of residence. Having looked at quite a few different definitions from different websites, it is all about the place where a person resides, some even describe it as a permanent or temporary place of residence. No reference whatsoever to ownership or legal right to live there.
Going with what was an uncontested definition of home from my quick search, if a man moves into a woman’s house that she owns/rents in her own name, the fact that he resides there makes it his home, by dictionary definition. As to any children he has, I’d say it depends if they spend enough time there to be regarded as living there. They should view it as their Dad’s home because it is, and I’d hope that they would feel able to be at home there even if it is not their main residence.
So yes, you can argue that if they mainly reside somewhere else, and their recorded address for things like doctors and schools is somewhere else, then no it’s not their home. However, it is their dads home, and they should be able to treat their dads home as their own. Frankly this whole thread is just an excuse to make kids feel unwelcome and unwanted.
If you are a woman who owns or rents a house on your own, do not have a man who has kids move in with you unless you are prepared to welcome them into your home as their home.