I'm not sure how universal this is, but in my area this is also about the point where you're loan repayments will become cheaper than renting the equivalent. And your loan repayments will only go down over time, rent will only go up.
I put down 5% and had to get PMI. But even after that my payments were less than renting an equivalent house. So 20% is definitely not a cut n dry rule.
Yes mortgage payment did include taxes, insurance, and PMI. (PMI is removed now, more on this below). But it doesn't include maintenance. I am paying almost $200 a month in HOA also which I didn't consider in my earlier comment. My guess is if HOA is considered then renting would be cheaper. (For context, I live in suburbs of Dallas).
Regarding PMI, my wife wanted to get our own house as soon as possible even if we had to pay PMI. I wanted to save 20% down first. It would have taken us more than a year to save 20% downpayment. So in end, wife won.
But that that time I didn't know that you could remove PMI as soon as you have 20% equity in the house. So we did our best, made extra payments, and built up equity to 20%, and removed PMI in about a year.
If we had tried to save for 20% downpayment like I wanted, we would have paid $12,000+ towards rent instead of about $1200 towards PMI (and a little more interests). So I tell all my friends that don't let PMI scare them, as long as they can afford to make extra payment and be able to remove it within a year or so.
For FHA backed mortgages the FHA changed the rules on PMI in 2013 where you are no longer able to remove the PMI after the loan reaches 78% loan-to-value and annual MIP has been paid for at least 60 months. For FHA mortgages finalized after June 3, 2013 the payments are there for the life of the loan. So can be worth considering refinancing once you are in the position to qualify for conventional financing and pay 20% down.
In general if you put down less than 20%, you have to pay for mortgage insurance, which pays neither principal nor interest. That may vary by region and by other things like first-time homebuyer or veteran status, so YMMV. But from first-hand experience I can say that paying 20% makes life a lot happier.
I'm not sure how universal this is, but in my area this is also about the point where you're loan repayments will become cheaper than renting the equivalent. And your loan repayments will only go down over time, rent will only go up.