Among the finite dimensional Lie algebras over a field of characteristic 0, there is a sensible definition of "reductive Lie algebra" going back at least to the 1960 first chapter of N. Bourbaki's treatise Groupes et algebres de Lie ($\S6$, no. 4 in the first printing, which I have at hand). Here the concise definition just requires that $\mathfrak{g}$ have a completely reducible ("semisimple") adjoint representation. This is immediately shown to be equivalent to a number of other conditions, e.g., that the derived algebra is semisimple, or that the nilpotent radical is zero, or that the radical of $\mathbb{g}$ is the center.
Is there a reasonable definition of "reductive Lie algebra" over a field of characteristic $p>0$?
For example, the undefined term occurs in the title of a 2000 paper by J.C. Jantzen here, though the text suggests right away that he is referring to the Lie algebra of a (say connected) reductive algebraic group $G$.
The current widespread use of the term "reductive" is sometimes problematic though well-motivated by the inductive role of Levi subgroups of parabolic subgroups in a given semisimple $G$. It reflects the influence of the 1965 paper by Borel-Tits here. For affine algebraic groups (or group schemes) the notion "reductive" just means that the unipotent radical of the group is trivial. The structure theory then shows that such a (connected) algebraic group is an almost-direct product of a semisimple group and an algebraic torus. This of course relies on the intrinsic nature of the Jordan-Chevalley decomposition, which is missing from the classical theory of Lie groups. There one often focuses on semisimple groups, having found (thanks to Levi and Mal'cev) that an arbitrary connected Lie group is a semidirect product of a semisimple Lie group (unique up to conjugacy) and a solvable Lie group (the radical).
The term "reductive" does have the drawback of suggesting complete reducibility of finite dimensional representations ("linear reductivity"). In characteristic 0, Mostow had already studied carefully the Lie groups which satisfy such complete reducibility for relevant finite dimensional representations. This property carries over easily to semisimple (hence also reductive) algebraic groups thanks to Chevalley's classification combined with Weyl's classical theorem for Lie algebras. But linear reductivity fails badly for reductive algebraic groups in prime characteristic. Here there is a weaker substitute called "geometric reductivity" conjectured by Mumford to hold for all reductive groups and eventually proved by Haboush. In any case, the Bourbaki definition of "reductive Lie algebra" isn't helpful in characteristic $p$. And the Lie algebra of a reductive group sometimes has a quirky structure, as seen already for $\mathfrak{gl}_n$ or $\mathfrak{sl}_n$ when $p | n$. It's definitely not easy to characterize such Lie algebras within the much larger class of restricted Lie algebras.
