What exactly is captured in a Sample of the Nyquist-Theorem variety?
It might be better to think of Nyquist sampling as shooting fish in a bucket.

The bucket limits what is in the bucket by being only so large. This is like a band limit. We can only have so much variation.

Two samples is the very least number of samples we need to specify that 1 sec of 1 Hz sine wave, although intuition tells us we are on the edge.  Three samples for a second and half is a safer proposition.   How about 120 samples for 60 seconds of 1 Hz signal. If you look at the reconstruction formula for Nyquist sampling, it says you need a need an infinite number of samples to get that 1  Hz sine back.  The summation limits are plus and minus infinity.

In practice, we usually sample a bit faster than what Nyquist specifies. 

There are circumstances where one can sample less than what Nyquist tells us and there is a lot of activity in an area known as sparse sampling. 

To be honest, this was a terrible analogy. The mathematics is really more straightforward. Things like mathematical limits are hard to analogize.

It also turns out that a 1Hz sine wave is also a mathematical ideal.