The first, as well as I understand, is a standard definition for 2-parametric Pareto distribution. Two of the recommended manuals for exam P - Hassett & Stewart, and Wackerly, Mendenhall, & Scheaffer - use the definition.
The second definition is used by Bean.
The three other recommended manuals describe 1-parameter Pareto distribution only, using definition 1 with x0=1.
The expected values calculated according to definitions 1 and 2 differ by x0. For example, in the following mock exam problem
Two different "right" answers are possible, depending on definition of 2-parameter Pareto distribution. "B" and "D" ($250 and $750). "B" if you used the Bean's book while preparing, or "D" if you read instead Hassett & Steward or many other manuals.
The Bean's definition can be used to describe insurance payments after deductibles, while the more standard definition describes the loss, including deductibles. In substance this is the same information. Just need to understand what is expected during exam.
Your #1 is for a single parameter Pareto.
Your #2 is for a two parameter Pareto. Unless they state "single parameter Pareto" use #2 for F(x). Your original posting is in error. It's been a while since I've sat for P, so I'm not familiar with those textbooks and how they explain the distributions. For a complete listing of distributions (You don't have to memorize them all) download the tables from exam M.
The link is: