$$ Y=X\theta+\varepsilon $$ is a linear model which splitts of the zero-mean stochastic part $\varepsilon$ form the deterministic part $X\theta$ of $Y$. It is not generally true that $\hat Y=X\theta+\varepsilon$. Usually, we take $\hat Y=X\theta$, and this is an optimal point prediction under square loss.

$$ Y=X\theta+\varepsilon $$ is a linear model which splits off the zero-mean stochastic part $\varepsilon$ form the deterministic part $X\theta$ of $Y$. It is not generally true that $\hat Y=X\theta+\varepsilon$. Usually, we take $\hat Y=X\theta$, and this is an optimal point prediction under square loss.