p. 152-153 of "Using Algebraic Geometry" (2nd. ed) by Cox, Little, and O'Shea says:
[S]uppose, that a collection of $n$ polynomials $f_1, \dotsc, f_n$ has a single zero in $k^n$, which we may take to be the origin. Let $I = \langle f_1, \dotsc, f_n \rangle$. [O]ne can show that most small perturbations of $f_1, \dotsc, f_n$ result in a system of equations with distinct zeroes, each of which has multiplicity one, and that the number of such zeroes is precisely equal to the multiplicity of the origin as a solution of $f_1 = \dotsb = f_n = 0$. Moreover, the ring $k[x_1, \dotsc, x_n]/I$ turns out to be a limit, in a rather precise sense, of the set of functions on these distinct zeroes.
I'm interested in any sources that discuss how to define precisely how $k[x_1, \dotsc, x_n]/I$ is a limit in the above sense, but unfortunately, the text doesn't mention any additional sources.
More broadly, I'm interested in any relatively modern treatments of a dynamic definition for intersection multiplicity, which is also alluded to in this answer. Unsurprisingly, the static definition (in terms of the dimension of a local ring) is much cleaner, so all the sources I've seen take that as the definition, and maybe mention that it's related to how perturbing the curves around an intersection point of multiplicity $m > 1$ will result in $m$ points of multiplicity $1$, but no one bothers to explain it precisely.
There are also some historical sources, explaining how there used to be a principle called "conservation of number" which describes the above, but that comes from the old Italian school of algebraic geometry, and those sources are probably too unrigorous to be useful. It seems that Cox, et al. knew of at least one source that treats it in somewhat modern language.