There is no ambiguity or speculation regarding the usage of completeness for the authors didn't resort to any creative nomenclature but rather stuck to the conventional one.
In functional analysis, there is the general concept of totality: a subset $M$ in a normed space $X$ is total if and only if $\overline{\operatorname{span}~ M}=X.$ It can be shown that (cf.$\rm[I]$) that if $X$ is complete, then $M$ is total if for any $x\in X, ~~\ x\perp M\implies x=0.$
As noted in this lecture, the same can be seen here:
This use of the word complete is analogous to calling a set of vectors $ v_1, \ldots, v_n$ complete if they span the whole space, that is, any $v$ can be written as a linear combination $v = \sum a_jv_j$ of these vectors. This is equivalent to the condition that if $w$ is orthogonal to all $v_j$’s, then $w =0.$ To make the connection with the [definition] let’s consider the discrete case. Then completeness means that $g(t)P(T = t;\theta) = 0 $ implies that $g(t) = 0.$ Since the sum may be viewed as the scalar product of the vectors $(g(t_1),g(t_2),\ldots) $and $(p(t_1),p(t_2),...)$, with $p(t) = P(T = t), $ this is the analog of the orthogonality condition just discussed.
For the sake of completeness, Lehmann and Scheffé used the term complete kernel in their $1947$ paper to introduce the concept.
Reference:
$\rm [I]$ Introductory Functional Analysis with Applications, Erwin Kreyszig, John Wiley & Sons., $1978, $ sec. $3.6, $ p. $169.$
