(I also have dreams where I wake up over and over and over, only to realize that shit, I'm still dreaming. It's surprisingly disconcerting, there's a kind of physical sense of losing my mind involved.)
In any dream state, you can test the rules of the world around you and see that your brain doesn't quite build the world to completeness. Light switches will not work, text will morph and shift either in front of your eyes or when you look away and turn back, the people around you will spontaneously morph into others, you can breathe through your nose when it's plugged, etc.
What happened to you is referred to as a false awakening. Testing the reality of your situation by checking clocks, breathing through your nose, counting your fingers, etc. will all quickly put you into a more lucid and aware state. From there you either continue or, more commonly, you wake up.
I haven't thought of that. What became my anchor for the past few years have been my schedule or being aware of it.
But there were rare cases when in my dream state, the reality is happening in the dream and that totally puts me off greatly. A good example is when I'm dreaming I woke up, went to work, and got home.
When I woke up, I didn't know until I arrived at the office and learned no day has passed yet.