[...] the question of what counts as a clock.

In order to define the notion of "clock" in general, in the context of (experimental) physics, it is instructive first to note certain more specific types of "clock" (whose explicit operational definitions may be more familiar already), namely

• good clock (cmp. MTW Figure 1.9) as a participant "$A$" (given as ordered set of his/her/its indications: $A \equiv \{ A_{\alpha} \}$) with a real-valued parametrization $\tau : A \rightarrow {\mathbb R}$ such that

  $ \forall A_{\alpha}, A_{\beta}, A_{\lambda} \in A \, {\rm with} \, A_{\alpha} \prec A_{\beta} \prec A_{\lambda}:$ $$ (\tau[ \, A_{\beta} \, ] - \tau[ \, A_{\alpha} \, ]) {\rm Duration}[ \, A_{\beta}, \, A_{\lambda} \, ] == (\tau[ \, A_{\lambda} \, ] - \tau[ \, A_{\beta} \, ]) {\rm Duration}[ \, A_{\alpha}, \, A_{\beta} \, ];$$ and

• ideal clock (cmp. MTW Box 16.4) as a participant "$A$" (given as ordered set of his/her/its indications: $A \equiv \{ A_{\alpha} \}$) of which there is a subset of "ticks" $T \subseteq A$ identified such that

  (1) each "tick" can be indexed by an integer: $\exists f: T \rightarrow {\mathbb Z}$, where

  - the indexing of "<i>ticks</i>" is distinctive:
  
    $\forall A_{\alpha}, A_{\beta} \in T \, :$ 
    $$(f[ \, A_{\alpha} \, ] == f[ \, A_{\beta} \, ]) \iff (A_{\alpha} \equiv A_{\beta})$$,
  
  - the indexing of "<i>ticks</i>" by integers is (strictly) monotonous to their order:
  
    $\forall A_{\alpha}, A_{\beta}, A_{\lambda} \in T \, :$ 
    $$(A_{\alpha} \prec A_{\beta} \prec A_{\lambda}) \iff (f[ \, A_{\alpha} \, ] - f[ \, A_{\beta} \, ]) * (f[ \, A_{\beta} \, ] - f[ \, A_{\lambda} \, ]) > 0$$,
  
  - the indexing is without gaps:
  
    $\forall A_{\alpha}, A_{\lambda} \in T \,:$ 
    $$\forall k \in {\mathbb Z} | (f[ \, A_{\alpha} \, ] - k) * (k - f[ \, A_{\lambda} \, ]) > 0 \implies \exists A_{\beta} \in T | f[ \, A_{\beta} \, ] = k$$; and
  

  (2) the indexing of "ticks" by integers constitutes a "good clock":

  $ \forall A_{\alpha}, A_{\beta}, A_{\lambda} \in T \, {\rm with} \, A_{\alpha} \prec A_{\beta} \prec A_{\lambda} \,:$ $$ (f[ \, A_{\beta} \, ] - f[ \, A_{\alpha} \, ]) {\rm Duration}[ \, A_{\beta}, \, A_{\lambda} \, ] == (f[ \, A_{\lambda} \, ] - f[ \, A_{\beta} \, ]) {\rm Duration}[ \, A_{\alpha}, \, A_{\beta} \, ].$$

Prerequisits are of course the notions of "distinguishability" and "ordering" of the indications of any suitable participant (for which there may not be given any further definition, but which are instead rather a prerequisit in asking for any definition); and the notion of "duration" (or at least: "duration" ratios) for which the presciption of MTW Box 16.4 may itself be considered an operational definition (provided the necessary construction of "Schild's ladder" and identification of "free falling particles" could be followed without requiring of "duration" ratios in turn already).

Consequently, any "clock" in general, which is characterized neither as "ideal" nor "good", would have to be defined by even weaker requirements; suitably perhaps as a participant "$A$" (given as ordered set of his/her/its indications: $A \equiv { A_{\alpha} }$) with a real-valued parametrization $t : A \rightarrow {\mathbb R}$ such that $t$ is a (strictly) monotonous function:

$$\forall A_{\alpha}, A_{\beta}, A_{\lambda} \in A | (A_{\alpha} \prec A_{\beta} \prec A_{\lambda}) \iff (t[ \, A_{\alpha} \, ] - t[ \, A_{\beta} \, ]) * (t[ \, A_{\beta} \, ] - t[ \, A_{\lambda} \, ]) > 0$$.

Examples of such general clocks are the various "forensic clocks", cmp. https://www.google.de/search?q=%22forensic+clock%22 which are not characterized by any particular "ticks" or "regularities", and which are not "good" by definition (but which may be found only incidentally more or less "good" by comparison with a "good clock" or an "ideal clock"; i.e. by measuring their "goodness" a.k.a. "accuracy").

Clearly, it's a measuring instrument

Clearly, of any given clock except an "ideal clock" (according to the prescription of MTW Box 16.4, as far as it can be followed at all) it can be asked and measured, whether it was "good" (a.k.a. "accurate"), or how it deviated from having been so, in any particular trail. Therefore the defining and primary measuring instruments or setups are only any "ideal clocks"; while all other "clocks" would at best depend on a "calibration chain" and/or be subject to corresponding assumptions.