He wants or he wanted

Both are grammatically correct.

#1 wanted is simple past tense

#2 wants is present tense

Neither are correct. It should be This morning I TALKED with Peter and he told me that he WANTED to change jobs, or This morning I TALKED with Peter and he told me that he WANTS to change jobs.

Having said that there is a subtle difference between the two, the first one means that at the time of asking Peter wanted to change jobs, that may still be the case, but things may have changed. Peter might have been given a raise or promotion since being asked and no longer wants to move.

In the second version he still wants to change jobs.

What Izzy is pushing is largely typical Izzy no-nothing bullshit. Peter might have died, he might have inherited a fortune, he might have blah blah blah, all of which is completely useless speculation.

The two sentences mean the same thing, they are examples of reported speech wherein a native speaker has the option to report in either the present or the past.