There may be misunderstandings here. First of all, you were right to initially think it's white's turn, because that's the diagram I provided. So that's entirely understandable.
Secondly, my assumption is that the side to move isn't relevant actually - right now. Notice that from the diagram in the book itself, two paragraphs are devoted to ideas: one for black and one for white. Assuming the same diagram! So the side to move appears to be irrelevant within the scope of providing ideas "in general" as the author seems to be doing.
Sorry about having taken things off track then.
Going back to your original question, the meaning of "As for white, he can determine the pawn structure by either pushing d4-d5 or playing dxe5, the latter only making sense if Nf3 -> Nh5 -> Nf5 can follow.", I think it's basically just saying those are options for white if the conditions are favorable. The author then gives an example in the excerpt you provided of where dxe5 works out because black misplayed, and @yetanotheraoc linked to one in his post. It's basically just making you aware of the idea.
There may be misunderstandings here. First of all, you were right to initially think it's white's turn, because that's the diagram I provided. So that's entirely understandable.
Secondly, my assumption is that the side to move isn't relevant actually - right now. Notice that from the diagram in the book itself, two paragraphs are devoted to ideas: one for black and one for white. Assuming the same diagram! So the side to move appears to be irrelevant within the scope of providing ideas "in general" as the author seems to be doing.