I'm a U.S. native. It's perfectly correct.
You need to UNDERSTAND the first three forms. I don't think there's any need to USE them. The second set of forms are just as natural and may even be easier to understand.
With the advent of digital watches I think it is now much more common to use the second form. It comes naturally when you are reading from a digital watch or clock--just as it is more and more common to give time to the minute without rounding.
If you ask me "what time is it?" I am going to say "Eleven forty-six" because that's what I'm seeing on my computer screen. I am not going to think about it, round to a quarter hour, and "translate" to "a quarter of twelve," I'm just going to read off what I see.
In the old days, when you were looking at an analog clock, you didn't expect it to be accurate to the minute, and you tended to think of time in terms of the position of the hands. Not today.