You cannot tell who caused the flare
Summary: The act of using a class feature is not observable unless the feature has text that makes it clear it is. Because Warding Flare has no such language, you cannot tell who is causing it.
The exception to "no hidden rules" is the interaction with everyday things. The game assumes that interactions with mundane things work like in real life, without spelling it out every time -- so you cannot walk through solid walls, even though there is no rule that says so, etc.
However, class features that cause supernatural effects are not like that, they behave like spells, where the rule is that they only do what they explicitly say. And Warding Flare is such a feature.
Some such class features say that they have observable components that make it obvious who is causing them. One example is the bard's Bardic Inspiration which says: You can inspire others through stirring words or music.
Others tell you under what circumstances their use has observable components that make it obvious who is causing them. For Spellcasting it is only possible to know who cast the spell, if there are observable components. Xanathar's Guide to Everything's section Perceiving a Caster at Work states:
If the need for a spell’s components has been removed by a special ability, such as the sorcerer’s Subtle Spell feature or the Innate Spellcasting trait possessed by many creatures, the casting of the spell is imperceptible. If an imperceptible casting produces a perceptible effect, it’s normally impossible to determine who cast the spell in the absence of other evidence.
Still other features, like Warding Flare, don't have any such language. For those, because there are no hidden rules, and there is no rule that says you know when someone is using a class feature, you cannot tell who caused them. The light domains Warding Flare just says:
When you are attacked by a creature within 30 feet of you that you can see, you can use your reaction to impose disadvantage on the attack roll, causing light to flare before the attacker before it hits or misses.
And Improved Flare does not add any visible actions, either. Both descriptions do not mention the cleric doing anything observable to make the light flare up, so there is nothing. Reactions are a construct of the action economy rules, not something inherently observable.
The effect of using the feature, the flare itself, is clearly observable, but the act of causing it here is not. Because there is no observable element of the act of using the feature, like when a spell has no observable components, it is not possible to tell who caused the flare.