what is difference between Alias vs. down sampling?
"Alias" is to "downsampling" what "a broken leg" is to "jumping from a 40-feet height". An effect from a cause, with a very probable risk in general, but could be harmless if you are lucky.

I am not sure there is a clear shared difference between uses of downsampling and undersampling. To me, used as a verb, "to downsample" may suggest a rate reduction by an integer or a fraction. While "to undersample" could be more general (at some sample rate below its Nyquist rate). Some definitions equate undersampling and bandpass sampling.

Used as an adjective as in "the signal is downsampled" or "the signal is undersampled", I understand different meaning. In the first one, I suppose a voluntary action. In the second one, this can be interpreted as an observation: the signal has been sampled at a rate lower than expected.