The things you want to be looking at are the differences between regulated and unregulated power supplies, and their outputs 'under load'.
As long as whatever you are powering utilizes less than the rated maximum milliamps, a regulated 9v linear power supply or a switched mode psu will always output 9v. The purpose of the extra regulation circuitry inside the supply is to keep the output voltage steady regardless of the current draw. It will just get hotter the more current you use.
If you have an unregulated linear power supply, the voltage it provides will drop under load. As you've discovered, if you measure the output of a 9v unregulated supply while its not powering anything, it'll probably be outputting around 12-13v. Thats because theres no load on the supply.
As soon as you use it to power something, the output voltage will drop in inverse proportion to the current draw. Some unregulated supplies will only be outputting their rated voltage when whatever they are powering is pulling their maximum rated milliamps, although in reality there will be a certain safety margin.
Realistically its perfectly safe to use an unregulated supply with most devices, as most modern kit will have internal power regulation circuitry anyway. In this case you don't really need a regulated PSU as variations on input voltage don't really matter that much. A device thats supposed to run on 9v with internal regulation, will probably be quite happy running on 12v or probably a little more. The internal regulation will be taking it down to a steady 5v or so anyway regardless of the input voltage. It'll just get a little warm at higher voltages.
You only really need a regulated power supply with devices that have no internal regulation, or are very sensitive to the noise that some unregulated supplies produce.