Even the tendentiously syntactic CGEL agrees that "At the general level [i.e., cross-linguistically] it is quite legitimate to invoke meaning" (Ch 1, §4, General and language-particular definitions). There are many syntactic facts that might help you decide. In some cases, such as Dinka, which my friend Bert Remijsen studies, phonology, even tone, may provide clues.
If I were looking at Old English and trying to decide if it has a specifier function, here are the things I think I'd start looking for:
a. Almost certainly a dependent in the NP
b. Likely precedes the head, but no a priory reason that it must.
c. Should have an identifiable position relative to other dependents within the NP.
i. This may vary for different constructions. (This could become circular if “construction” is defined based on the position of the assumed specifier.)
2. Constituent structure.
a. The specifier may have a head + dependent(s) structure or may be single-branching.
3. Participation in various constructions. (This may confound with position, above)
a. May participate in various constructions such as
ii. poss. + dem. + adj. + noun (Traugott 1992:173)
a. Almost certainly unique in its relationship to the head of the NP (Nom?).
5. Result of omission
a. May have no significant effect on semantics, pragmatics, or grammaticality.
b. May result in different semantics or pragmatics, but retain grammaticality.
c. May result in ungrammatical NP.
d. It’s possible that each of the above is true in the right construction.
6. Inflectional morphology.
a. Expected to agree with head noun where possible.
7. Alignment with various semantic roles (e.g., agent)
a. “The function underlying all determiners in Present-Day English is to mark the identifiability status of the referent” (Breban 2012:272).
b. Mark definiteness
c. Mark number
8. Fulfillment by members of particular categories (strong potential for circular definitions)
a. Genitive NPs
d. Interrogative and relative words
e. Perhaps a D category
f. Perhaps Adj
On top of that, I'd also consider the results of:
10. Psychological tests (e.g., reaction time)
11. Theoretical Mathematical Models (e.g., ?)
12. Empirical Computational Models (e.g., learnability)
Anything you might be able to add to this quick sketch would be most gratefully welcomed.