Different names for model with parameters specified and not? Say I have a general model:
$$y=\beta_{1}x_{1}+\beta_{0}$$
or
$$y=\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{0}$$
I might be performing some operations to determine which general model to choose.
Say I have decided on a general model structure and want to define a specialised model, like:
$$y=5 x_{1} +2$$
What is the vocabulary to call (1) model without parameter values defined (2) model with specific parameter values (can this even be called a model, are names function or equation more appropriate)?

The coefficients $\beta_1,\beta_2,\beta_0$ in this context are often called parameters.  Particular values of the parameters determine a particular member of a parametrized family of models.
In the context of linear regression, one sometimes says those three parameters are fixed (as opposed to random) and unobservable, so that they must be estimated by using least squares.  The variables $x_1,\,x_2$ are typically observable and often treated as fixed rather than random, even though they may change when a new sample is taken.  The rationale for treating them as not random is that one may be concerned with the conditional distribution of $y$, or of the least-squares estimates of the parameters, given the $x$-values.  On the other hand, $y$, although observable, would be treated as random.
