I think the author actually adds to the confusion about TSP by being unnecessarily vague.
Both P and NP are classes of decision problems. That means they only contain problems that ask for a yes/no answer. So because TSP asks for a route, it is not just a decision problem, and can't be in NP. However, as the author mentions, NP does in fact contain the TSP-DECIDE problem in the way he defines it.
Moreover, the problem TSP of finding a lowest cost route can be solved by a polynomial time Turing Machine that can use a TSP-DECIDE oracle (i.e. call a constant time function that solves TSP-DECIDE). It is in fact complete for that class FP^NP ("= function problems solvable in polynomial time with an oracle for NP problems"), which means it is one of the hardest problems there. The class FP^NP contains all of FNP (NP for functional problems) and is believed to contain more problems -- so there is a good chance that TSP is not in FNP.
Both P and NP are classes of decision problems. That means they only contain problems that ask for a yes/no answer. So because TSP asks for a route, it is not just a decision problem, and can't be in NP. However, as the author mentions, NP does in fact contain the TSP-DECIDE problem in the way he defines it.
Moreover, the problem TSP of finding a lowest cost route can be solved by a polynomial time Turing Machine that can use a TSP-DECIDE oracle (i.e. call a constant time function that solves TSP-DECIDE). It is in fact complete for that class FP^NP ("= function problems solvable in polynomial time with an oracle for NP problems"), which means it is one of the hardest problems there. The class FP^NP contains all of FNP (NP for functional problems) and is believed to contain more problems -- so there is a good chance that TSP is not in FNP.