When I extend my flaps to 10 degrees, what exactly is the 10 degrees measuring? Is this referring to the angle of the flap blades themselves, the new angle of the wing chord, the change in the new critical angle of attack or something else?
The "Degrees" of flap deployment generally refers to the deflection from the neutral (0° - flush with the rest of the wing) position of the flaps.
For most GA aircraft this is an actual angular measurement - 10° of flaps corresponds to the flaps lowering 10° from their neutral position, but the degree of travel can also be given as a percentage (0-100%). The aircraft's maintenance manual will specify the actual travel (and specify how it is to be measured), and if specific position detents are provided there will usually be "rigging limits" specified for where each position stop must be & how accurate the control position must be at each stop.