Output in a half wave rectifier

Yes, it would be no DC in a strict sense. If the resistor is ohmic and the diode ideal, the current would look like a sine wave, where one half is set to zero. Though it would not be an AC, if AC is interpreted as strictly sinusoidal (with full wave). Sometimes this kind of waveform is called "pulsed DC", especially if a smoothing electrolyte capacitor is put in parallel to the resistor.

A current does not flow simply because you make a connection; you have to make two connections to form a circuit. When the diode is reverse biased it is like a switch being open, current cannot flow. When the diode is forward biased it turns on to complete your circuit.

The output through the load looks like this:

The diode blocks one polarity of the AC cycle but allows the other polarity through.