Although a reasonable part of the question is about the commonly-accepted senses of the terminology, another part of it is surely about facts (rather than terminological conventions).
And I am sympathetic to bad reactions to the (fluctuating per source, and over time!) terminology. E.g., in other contexts, "symmetric" (over $\mathbb C$) would mean $\mathbb C$-bilinear, while "hermitian" would mean ... um, hermitian? ($\mathbb C$-sesquilinear?).
Ok, we can agree that the terminology is not self-explanatory? In my experience, disparities among usage and implied properties are as much due to differences in conception of (apparently official, but in different ways in different peoples' minds) what these words mean.
Nevertheless, yes, there are actual mathematical questions here that do not depend on what we declare (or, worse, tacitly presume) words to mean.
So, for example, "symmetric" does not refer to the complex-bilinear versus complex-sesquilinear issue, but to the issue of, if the operator is unbounded (hence, not defined, nor definable everywhere, by the closed graph theorem), whether or not it "appears to be self-adjoint" on its domain.
And there's the important-yet-hard-to-promote issue of whether some random unbounded operator is densely defined. Things tend to screw up if not. E.g., no adjoint. But/and giving things a name that does not refer to this feature is ... unhelpful. So, really, we should say that we care about densely-defined, symmetric operators. (Or maybe call them "potentially self-adjoint", though there's the problem that (cf. von Neumann) not all symmetric operators have self-adjoint extensions!)
E.g., sure, continuous (a.k.a. "bounded") operators don't have any of these issues... In any reasonable sense, symmetric=hermitian=self-adjoint.
For genuinely unbounded operators, symmetric does not imply self-adjoint, and, unless the thing is already self-adjoint, its adjoint is definitely not symmetric. (Crazy, right?)
But, in practical situations, all these seeming bait-and-switch or faux-paradoxical things are actually sensible. Abstractly, it's hard to see the forest for the trees, or vice-versa. In my opinion, this is yet another instance where too-aggressive generalization/abstraction makes things incomprehensible, if it goes so far as to disconnect from the (oh-so-) tangible examples that gave rise to the ideas.