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Hi exnihiloest,

What you propose could be possible but in this particular case we should see some sinusoid voltage waveform at the output instead of the still uggly pulse shape.  LC resonance always involves forming sinusoid or distorted sinusoid waves but not such straight lined pulse shapes.
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You are generally right, especially at low frequencies (< 1 Mhz). Nevertheless there are often different modes of resonance at very different and not harmonic frequencies (for example due to capacities between near turns but also between successive layers, or due to L if the wounding is built in several separated blocks...). This is particularly true at HF frequencies, because we are no more in the quasi-stationary regime, there is propagation effect along the wire (the current cannot be considered constant in the whole circuit).
To be sure that the observed signal is not due to such a phenomenon, the only way is to use a sin generator and to scan frequencies up to 20 Mhz. If there is not only one resonant frequency, the observed "ugly" signal is probably due to HF components of the pulse that make "ringing" the coil.