I like that definition, however there are ships which don't fit those specifications yet would also be considered (or not considered) aegis.
For instance, there are ships with PARs and VLS that are more designed for "shorter range" AAW, such as the thales APAR equipped saschen and de provincien class frigates, or as you state, the FCS-3A equipped akizuki. All such ships have relatively lower range PARs (effective range out to 200km as compared to the longer 400km+ ranges of SPY-1, SAMPSON, type 348 etc), and seem to emphasize medium range air defence more than longer range air defence.
However, they are all capable of decent missile saturation defence at mid range as well -- namely due to their radar's ability to simultaneously illuminate more targets than say, aegis ships which require mechanically mounted illuminators, I think.
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Another example is the Indian Kolkata class, which is equipped with the 250km+ MF STAR S band APAR, which is probably comparable to the SPY-1 in at least range. So, its radar has competent range performance but its AAW relies on the 70km barak 8. Assuming that 70km figure optimistically reflects its maximum effective range, it is still more of a medium-long range SAM rather than true long range SAM like SM-2, Aster 30, HQ-9. Would the Kolkata thus be considered "aegis-like" or not?
(Note, the Barak 8 is an active radar guided SAM, so it should have better multi target engagement than a ship equipped with only SARH guided missiles)
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Another example we can include are the Kirov and Slava classes.
On paper they have highly capable missile systems with naval S-300s that can reach out to nearly two hundred kilometers. However we don't know how well they can handle saturation attacks, and their is skepticism regarding the soviet era combat management systems and electronics, I think.
So, aegis-like, yes/no?
I think a ship's "aegis-like"-ness can effectively be rated on a series of continuous (rather than categorical) scales, measuring domains such as air defence, surface attack, land attack, ASW.
There will be no "overall" score that seeks to unify the above domains, because every ship's capability will differ depending on its mission and loadout available (for instance, the JMSDF's aegis ships lack cruise missiles, but that shouldn't necessarily decrease its score compared to an aegis destroyer that can fire tomahawks).
However there are common scores we can attribute to various ships, no matter their "mission" or "loadout" -- combat management, damage resistance, cost, should all be relatively unbiased measures to compound upon the individual combat domains which every ship might excel in or lack in.
Maybe if we had a lot of time we could all chip in and debate a common metric for measuring the combat capability of various ships lol. We could call it the SDF warship competency scale or something.
But alas, the only conclusion we can maintain is that the "aegis" label is only one which is useful for marketing and showing off to the uninformed enthusiasts of the world
