Question about delayed sampled sinusoid math expression | Algorithm Design | Forum

I have been studying the digital audio processing by using the book .
According to the book at page 199, for analog Sinusoid:

Complex Sinusoid = e^(jwt)

Delayed Sinusoid = e^(jw(t-n))=e^(jwt) * e^(-jwn), a delay of n seconds

For digital sampled version:

sampled complex sinusoid = e^(jwnT), T is interval for each sample, n is the index of sample

I understand all above, but I got confused about the delayed sampled sinusoid which described as: e^(jw(nT - M)) , M = samples of delay

But I think it should be described as e^(jwT(n - M)), since the T is a constant for a fixed sample rate, n and M has the same unit.

At first I thought that maybe a typo, but the following computation parts of the book are all using the e^(jw(nT-M))as basis and drew some conclusion

Anyone can explain it for me?