Maybe the best answer I can give is ''learn about deformation theory''. But I will take a shot at writing the dictionary you request, in the case where $E(u)$ is a polynomial function
- linearization stability <--> smoothness / instability <--> singularity
- linearization stable/unstable point <--> singular / smooth point
- linearized solutions <--> Zariski tangent space
- functions like $Q[u]$ <--> obstructions
- the zero set $Q[u]=0$ <--> (Edited.) the tangent cone
- functions like $S_i[u]$ <--> the functions $\partial E/\partial u$, or maybe it is better to say, a certain Fitting ideal of the sheaf of Kahler differentials on $X$
- the zero set $S_i[u] = 0$ <--> on $X$, you would I guess call it the singular locus; in the whole ambient space I don't know.
Regarding ''learn about deformation theory'', the point is that you started with some very explicit algebraic variety $X$ and then decided to think about its structure near a point. But deformation theory lets you think about the following thing: say you want to think about the space of all algebraic ''solutions'' of some sort near a given ''solution'', but to a less explicit problem like ''describe all holomorphic maps from a genus 5 curve into projective space'', which of course can also be written as a PDE. It may be difficult or worse to construct such a space globally, but still you can start to think about the local infinitesimal structure near a given solution using infinitesimal methods; and then there are very powerful algebraic tools (the Artin approximation theorem) which let you under good conditions construct an algebraic neighborhood of your given solution.