Which definition of the integral is more "mathematically rigorus"?

Which one of these defintions is considered more "mathematically rigourous"? And which one is easier to manipulate? That is, which one is used more often to manipulate in a manner that proves certain theorems? Or, is there a different definition that is used? Also, what other definitions of the integral are there then these? (And please don't hesitate to point out any errors in the definitions below, because I'm no expert (Giggle))

[A]

[B]

Thanks in advance