To add another misconception, the Nyquist frequency is a lower bound, below which you necessarily get aliasing. It doesn't say anything about whether said sampling rate is sufficient for reconstruction or whatever your intended use is.

Yes, it does. The Nyquist criterion gives exactly the (minimum) sampling frequency you need for perfect reconstruction of a bandlimited signal.

E.g. sampling a 1hz signal at 2hz still doesn't tell you if the signal was a 1hz sin or a 1hz sawtooth (depending on how lucky or unlucky you are).

A 1 Hz sawtooth is not a bandlimited signal, so the Nyquist theorem does not apply.