Two different expressions for lead compensator?Why?
There is no standard.  People use the form they think works best in a particular situation, or what they were taught back in the day, or what they feels best makes the point of the lesson they're teaching.
The things that actually matter are the pole and zero location, and the overall gain (although you may end up expressing that gain at DC, at high frequencies, or even in rare circumstance along the rising amplitude vs. frequency slope).
Just get used to different authors saying the same things at different times.  Sometimes it's a pain -- sometimes it gives you more insight into a subject.