I've used this card in my deck in several hundred games and I've managed to get it special summoned exactly once.
Let's answer some questions: Gravekeeper? Nope. Keith's dragons? Nope. 4 starred ladybug? Nope. Any spell/trap effect? Nope.
I can't remember exactly which card triggered the effect; I think it was an insect type, but I might be wrong. It wasn't anything used in any of the common pvp decks and it very specifically said on the card "target and destroy one enemy monster" as its effect (I think it was it's on death effect).
Dragonic Knights' effect says "when the effect of an opponents monster is activated that targets and destroys a card(s) you control", so it should therefore activate if one card or more cards are destroyed, whether it be monster or spell/trap.
4 star doesn't target
Barrel Dragon Targets first, tosses coins 2nd, then destroys.
That 2nd phase prevents Dragonic's effect from applying as they need to be targeted, then destroyed. If his effect read "if" not "when" then it would've worked.
Dragonic Knight doesn't work with Barrel Dragon because it isn't guaranteed to destroy a monster. (The same reason you can't use Destruction Jammer on it.) The card needs to target and be guaranteed to destroy it for Dragonic Knight to be used. (Oh, Barrel Dragon's coin flip and destruction happen simultaneously anyway, not that that has any effect on a Quick Effect being chained to it.)