In a recent post Drozd's trichotomy theorem for finite-dimensional algebras over any field I asked whether the classical Drozd's trichotomy (finite, tame, wild representation type) for finite-dimensional algebras over algebraically closed fields holds over arbitrary fields. I received as an answer that for indecomposable hereditary finite-dimensional algebras the trichotomy holds over any field, and the notions of finite, tame and wild can be expressed in terms of a billinear form associated to the Grothendieck group of the algebra. I wonder now if there is a general notion of tameness for arbitrary finite-dimensional algebras over non-necessarily algebraically closed fields.
In general, a finite-dimensional algebra $A$ over an algebraically closed field $K$ is called of tame representation type if for any $d\in\mathbb{N}$ there is a finite number of $A$-$K[X]$-bimodules $M_1,...,M_n$ which are free of rank $d$ as right $K[X]$-modules s.t. all but a finite number of $A$-modules of length $d$ are of the form $M_i\otimes_{K[X]} K[X]/(X-\alpha_i)$ for some $1\leq i\leq n$, $\alpha_i\in K$.
I searched on the literature and on the references suggested in posts such as What is a good reference on the trichotomy theorem? and What is the simplest example of the tame representation type? in order to find a general definition of tameness that does not require the field to be algebraically closed, but in all of them it was defined only for algebraically closed fields. My feeling is that in this definition the tensor product with modules of the form $K[X]/(X-\alpha_i)$ is required in part because any simple $K[X]$-module is of that form as the only irreducible polynomials over $K[X]$ have the form $X-\alpha$, $\alpha\in K$. But over non-algebraically closed fields we may have irreducible polynomials not of that form, so I guess this definition may be extended by imposing some extra condition such that for example instead of just considering modules of the form $K[X]/(X-\alpha_i)$ we may need to consider all the possible simple $K[X]$-modules while tensoring.
Another idea I have is to say that an algebra $A$ is tame if $A\otimes_K \bar{K}$ is tame, where $\bar{K}$ denotes the algebraic closure of $K$.
Are any of these possibilities accurate for a general definition?
In case there is a well-established general definition of tame type I would like to ask, if possible, for a reference for it in the literature.