So does it classify as preamp?
A preamp is not a classification. It is a circuit function block you can make. You can make them using, amongst other things, with an operational amplifier.
An operation amplifier is normally defined as an amplifier with a differential input ( + and - ) and a single ended output. It has an extremely high ( unusable high ) open loop gain and to get the gain to be a usable value you apply negative feedback. The output impedance is normally in the range of 1K to 10K.
I need to clear up the impedance matching thing
Yes it is the source of all your misconceptions.
Ever thought why you can't turn over a car engine with 8 AA batteries? After all they can provide 12V just like your car battery. You know that you can't but why? Well it is a lot to do with the chemistry in a cell but all that, and more, gets bundled up in the concept of output impedance. It is the equivalent of a series resistor in line with the voltage generator. It forms a voltage divider with your load and if this theoretical resistor is several orders of magnitude greater than your load the potential divider action insures that there is "stuff all" voltage across your load.
So it seems weird the TL071 can drive larger impedance loads but not a 32Ω headset.
Any voltage source can drive a high impedance because as long as the load is comparable with the output impedance it will deliver a voltage to it. When the load is the same as the output impedance you get half the voltage across the load as the voltage generator produces.
This link is about Raspberry Pi GPIO output impedance but the principals apply to any voltage generator, battery, amplifier, output pin, transducer.
http://www.thebox.myzen.co.uk/Raspberry/Understanding_Outputs.html