It can be confusing. Just remember there is a diode from base to collector, and it should be reversed biased for active mode. As soon as it becomes forward biased, the transistor is in saturation.

 

As others have said, saturation is defined as when the CB junction is forward biased. Just exactly what the voltage is, is hard to define. It depends on collector current, and junction area, and temperature, and the phase of the moon.

The datasheet TELLS you how hard to drive the base to ensure saturation.
See the attachment.

 

I think that I am on the verge of understanding this. I looked at the datasheet and compared it to my voltage readings. I am going to sleep on it, and maybe tomorrow morning, it will be clearer. Believe it or not, sometimes I do wake up with a clear solution to a problem I couldn't fathom the day before.

Of course sometimes, I wake up and have forgotten the problem.