Again, it can be both.
Also, it's kind of tricky to argue that the purpose of Exigents was to broaden the definition of Exalted while simultaneously arguing that they didn't dilute the definition of the term.
I wouldn't say 'purpose' so much as 'consequence'.
The Exalted are the Chosen of the Gods. The most famous and iconic of their number, and the ones which consequentially feature in the core rulebook, are the Solar Exalted, who were Chosen by the King of the Gods. Similarly you have the Lunars and the Sidereals, chosen by Luna and the Five Maidens respectively. As the Chosen of the Incarnae, those three types of Exalted share certain similarities with one another, such as their limited numbers and the way in which they reincarnate rather than reproducing.
This is, in essence, the 'basic template' - it's what people mean when they refer to the 'autonomous titan-killing super weapons' and other terms of that nature.
But the Incarnae are not the only powerful beings who choose Exalted. They never have been.
The Terrestrial Exalted are the Chosen of Gaia, one step removed through her own children, the Five Elemental Dragons. They differ from that basic template by being able to multiply, passing their own power down through the blood.
The Infernal Exalted are the Chosen of the Yozis, and differ from the basic template by being corrupted versions of other powerful agents rather than created from scratch. The term was always applied to the Akuma as well as the more iconic Green Sun Princes, after all, and their themes overlap with but are distinct from the more traditional 'Exalted' template.
The Alchemical Exalted are the Chosen of Autocthon, and differ from the basic template by being
made by the humans he educated, rather than being chosen as individuals. They differ the most from the basic template, which is somewhat ironic considering the role that Autocthon is generally held to have played in the creation of the Exalted in the first place, but they are still deserving of the term.
I could go on, but hopefully you get the point.
What, then, do we call individuals who are chosen by gods but
not by the Incarnae? Certainly they exist, as a relatively minor but still well established part of the setting for several editions. Do we stick with God-Blooded, even when they don't actually have any direct blood relation to their patron in the same way that the Dragon-Blooded do, and keep using that same term for individuals who
are directly descended from a deity?
The answer that the designers chose was 'no'. They chose to call them 'Exigents'.
Because for all we might talk about how this new addition dilutes the significance of the name or diminishes the narrative significance of the non-Exalted players of Creation, in the end? This approach is
more consistent, more unified than what came before, not less.
Except this rules out Alchemicals as a class from being considered Exalted, and instead as Godblooded, because the mortals of Autochthonia are his Chosen People in the Biblical sense, and are Championed by their Exalts, who the Great Maker does not choose or empower directly except as embodying the means of fueling their creation.
Well, yes. By what right do the Alchemicals get to call themselves 'Exalted'?
They aren't born or chosen, they are made. Their charms aren't techniques or expressions of great skill, but individual tools built into their body and capable of being swapped around. They aren't individuals who caught the eye of a higher power, but a collected mass of heroic souls fused together into a singular whole as part of their construction. They did not participate in the Primordial War or cast down the enemies of the gods. They don't even rely on the direct intervention of a powerful spirit to create, since the whole thing is done by well educated mortals.
And yet, they are Exalted. They are the Chosen of Autocthon, who reflect his themes and his ideals and serve to represent his interests in the world.
If the Alchemicals qualify as Exalted, then the Exigents most certainly do as well.