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Figure 1 looks interesting since it has both a time and frequency axis, when usually signals have either a time __or__ frequency axis.

Now I'm curious how the Fourier (?) transform of a signal at a __single__ given timepoint is even defined ...

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The concepts you are looking for are the short-time Fourier transform and spectrogram:

https://en.wikipedia.org/wiki/Short-time_Fourier_transform

https://en.wikipedia.org/wiki/Spectrogram

Jamessb already linked to the right terms. One thing to add is that there is always a tradeoff between time and frequency resolution on short time Fourier transforms. You just can’t have both. It’s always a somewhat unsatisfying tradeoff that still works well in practice.

I found this out when I was trying to turn an ordinary 5 dollar thrifted musical keyboard into a midi controller by plugging it into my PC, putting it on "sine wave" and using a Goertzel detector

The latency for detecting audio-frequency waves is quite bad

This also stymied my desire to put digital audio onto a vinyl record :( literally not enough bandwidth

By definition there is vinyl quality worth of bandwidth on the record.

True, honestly the real limiting factor was my signals and encoding lack of knowledge

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