I was wondering if the second condition already implies the first? I mean, are there situations where the first condition is violated even though the second condition is not? And if so, how are the forces in that situation non-conservative even if they satisfy the second condition?
I think that the first sentence has the purpose of avoiding those forces that are explicitly dependent on time or velocity. For particular forces that still depends on time the second sentece could be true (at a fixed time).
About the second sentence, I think it is tru that a conservative field can be derived from a scalar potential which depends only on the position r. The force is minus the gradient of the potential, so I think it is true that for a conservative field the second sentence implies the first. However one must also specifies that is time-independent and also speed-independent