Your problem is excessive damping and the reason is the shape of your glass. Wine glasses have thin walls, a wonderful resonating bell shape and a thin stem attachment that prevents the resonance from being coupled to the supporting structure. When you tap a wine glass it doesn't just make a pitched sound it actually rings for a noticeably long time. That is to say the ringing is very poorly damped. A heavier walled nearer to cylindrical tumbler with a broad base touching the counter may make a distinguishable frequency , but it is short lived and doesn't survive much longer than the off resonant frequencies so you don't get a pure tone. A "tink" rather than a ring. This is why you never see a musician playing water glasses use anything but bell shaped stem ware right down to the bass brandy snifter. Tumbler shapes are too highly damped to be made to resonate with any reasonable input. Sound, even amplified sound from a speaker, is too weak to cause resonance unless the resonator has a nice high q.
Get a wine glass, wet your finger and rub it around the edge. If you can make the glass sing it will work for you experiment. If you can't it won't. You will also see the resonant quality (q) in your captured Fourier transform. The wine glass will ring for a much longer time and the transform will resolve to a much purer tone. If you measure while running your finger around it will be a beautifully pure tone.
As for putting a spoon in the glass, it is important that the item be light. Try a straw. Also, no matter what anyone says, you can shatter a wine glass this way. Please be careful. Start at lower amplitudes, and wear safety glasses. Better yet, stay well back.
If you have any sort of a high speed camera, the first resonant mode vibration of a wine glass looks very impressive achieving amplitudes without breaking that are a significant fraction of the diameter of the glass. You won't believe glass can bend that far.
