There are two common formulations of linear regression. To focus on the concepts, I will abstract them somewhat. The mathematical description is a little more involved than the English description, so let's begin with the latter:
Linear regression is a model in which a response $Y$ is assumed to be random with a distribution determined by regressors $X$ via a linear map $\beta(X)$ and, possibly, by other parameters $\theta$.
In most cases the set of possible distributions is a location family with parameters $\alpha$ and $\theta$ and $\beta(X)$ gives the parameter $\alpha$. The archetypical example is ordinary regression in which the set of distributions is the Normal family $\mathcal{N}(\mu, \sigma)$ and $\mu=\beta(X)$ is a linear function of the regressors.
Because I have not yet described this mathematically, it's still an open question what kinds of mathematical objects $X$, $Y$, $\beta$, and $\theta$ refer to--and I believe that is the main issue in this thread. Although one can make various (equivalent) choices, most will be equivalent to, or special cases, of the following description.
Fixed regressors. The regressors are represented as real vectors $X\in\mathbb{R}^p$. The response is a random variable $Y:\Omega\to\mathbb{R}$ (where $\Omega$ is endowed with a sigma field and probability). The model is a function $f:\mathbb{R}\times\Theta\to M^d$ (or, if you like, a set of functions $\mathbb{R}\to M^d$ parameterized by $\Theta$). $M^d$ is a finite dimensional topological (usually second differentiable) submanifold (or submanifold-with-boundary) of dimension $d$ of the space of probability distributions. $f$ is usually taken to be continuous (or sufficiently differentiable). $\Theta\subset\mathbb{R}^{d-1}$ are the "nuisance parameters." It is supposed that the distribution of $Y$ is $f(\beta(X), \theta)$ for some unknown dual vector $\beta\in\mathbb{R}^{p*}$ (the "regression coefficients") and unknown $\theta\in\Theta$. We may write this $$Y \sim f(\beta(X), \theta).$$
Random regressors. The regressors and response are a $p+1$ dimensional vector-valued random variable $Z = (X,Y): \Omega^\prime \to \mathbb{R}^p \times \mathbb{R}$. The model $f$ is the same kind of object as before, but now it gives the conditional probability $$ Y|X \sim f(\beta(X), \theta).$$
The mathematical description is useless without some prescription telling how it is intended to be applied to data. In the fixed regressor case we conceive of $X$ as being specified by the experimenter. Thus it might help to view $\Omega$ as a product $\mathbb{R}^p\times \Omega^\prime$ endowed with a product sigma algebra. The experimenter determines $X$ and nature determines (some unknown, abstract) $\omega\in\Omega^\prime$. In the random regressor case, nature determines $\omega\in\Omega^\prime$, the $X$-component of the random variable $\pi_X(Z(\omega))$ determines $X$ (which is "observed"), and we now have an ordered pair $(X(\omega), \omega)) \in \Omega$ exactly as in the fixed regressor case.
The archetypical example of multiple linear regression (which I will express using standard notation for the objects rather than this more general one) is that $$f(\beta(X), \sigma)=\mathcal{N}(\beta(x), \sigma)$$ for some constant $\sigma \in \Theta = \mathbb{R}^{+}$. As $x$ varies throughout $\mathbb{R}^p$, its image differentiably traces out a one-dimensional subset--a curve--in the two-dimensional manifold of Normal distributions.
When--in any fashion whatsoever--$\beta$ is estimated as $\hat\beta$ and $\sigma$ as $\hat\sigma$, the value of $\hat\beta(x)$ is the predicted value for $x$--whether $x$ is controlled by the experimenter (case 1) or is observed by him (case 2). If we either set a value (case 1) or observe a realization (case 2) $x$ of $X$, then the response $Y$ associated with that $X$ is a random variable whose distribution is $\mathcal{N}(\beta(x), \sigma)$, which is unknown but estimated to be $\mathcal{N}(\hat\beta(x), \hat\sigma)$.
