Output signal in digital communication
Equation (1) in your question is a pure additive noise channel without any further distortion of the signal. Equation (2) takes a delay into account, but this is usually not relevant. A pure delay does not affect the quality of the communication link in terms of bit error rate. (I'm not talking about the annoying effect of delay in voice communication).

So the model in Equation (2) is often not relevant and can be replaced by Equation (1). A more general and useful model, which includes Eq. (2) as a special case, is

$$y(t)=(x*h)(t)+n(t)$$

where $h(t)$ is the impulse response of the channel, and $*$ denotes convolution. This model represents a linear time-invariant channel with additive noise. Eq. (2) is obtained as a special case with $h(t)=\delta(t-t_0)$.