Does the hold part in sample and hold turns the signal from discrete to continuous? I am trying to link sample and hold to discrete and continuous time domain.
Could someone please confirm:

When sampling but not holding for "an extended" period of time, you have discrete samples.

When sampling AND holding you lose the "discretion" and obtain an approximated continuous signal.

Is this correct?
The sample-and-hold circuit is a physical, analog electrical circuit and, by definition, its output is continuous.

The actual conversion to discrete time happens at the quantizer, not at the sampler. The output of the quantizer is a discrete sequence of numbers associated with time instants, so it can be said to be a discrete signal.