yes, that's right. You may be well off with a 9mm since there are cheap practice ammo and it is not difficult to manage. There are discussions on "stopping power" but I think its a bit theoretical. 9mm works fine and you can have it in a smaller package. .45 in a 1911 is also proven beyond doubt but is is a larger piece. A .45 in a small frame is tough.
What matters is that you feel comfortable and have practiced so much that you don't have to think when using it, no matter if it's in competition or in a protection situation(try to avoid the last one)
And seriously, pull the bullet and look at them: Theres a very small difference between 9 and 10 mm. If you want power beyond 9, 10, 40, 45, you really need to look at rifles or shotguns. And for home protection you cant beat the shotgun: It is safer for you and family, more intimidating for the bad guy, easier to handle in a stressful situation etc.
And with any dangerous equipment: Make sure that no unauthorized persons, kids, burglars, can get hold of it-that may cause tons of grief and limit freedom in the future.