As meaning “field of fractions of an integral domain that arises as the completion of a local ring with respect to its canonical valuation”.

The first meaning is not too serious (and is seldom if ever considered seriously), since usually a field $F$ will not uniquely determine a local subring giving rise to it, nor does this meaning imply any tight connection to local topological conditions such as local compactness. Under this interpretation, $\mathbb{Q}$ would be a “local field”, which is virtually unheard of.

The second meaning of a local field has more content, because the Cauchy completeness (with respect to an $\mathfrak{m}$-topology, where $\mathfrak{m}$ is the maximal ideal of some local ring) determines the local ring via the topology: the complement of $x$ such that $x^{-n}$ converges to $0$. The second meaning occurs in the literature, as in the famous text Corps Loceaux by Serre.

Relation to local fields in algebraic number theory

There is nontrivial intersection with the notion of local field as defined in algebraic number theory, since the nonarchimedean local fields as defined in algebraic number theory are conspicuous examples of this second meaning. Observe however that

The archimedean local fields in algebraic number theory $\mathbb{R}$, $\mathbb{C}$ do not arise this way;

is typically not compact (and its field of fractions is not locally compact). It is of course compact if each $R/m^n$ is finite with the discrete topology.