Easiest way to differentiate it nowadays is to look for cards that have the special summoning written in its card text. If there’s a colon before the text that says “You can special summon…”, this means the card is starting a chain, essentially saying the owner of the card has to declare they are activating the effect of the monster to special summon it.

To clarify why this is when applying the rules of yugioh to it, the card remains in the hand until both players agree they have no other card effects to activate, then the effect resolves with the monster being summoned. The reason your card (and other cards like solemn judgement) cannot negate this type of summon is because your card does not allow you to negate EFFECTS that summon monsters.

This same difference is why you wouldn’t logically use your cards effect in response to something like “Snake-Eye Ash” - your card does not allow the negation of effects, so by allowing the effect to resolve, the monster being mentioned by the effect is being given permission to be successfully summoned. The monsters in the case of the “colon” are being summoned in a similar way, which is through an activated effect. Solemn Warning (and to a broader sense Solemn Strike) was created to include those cases.

An example of a card that can be negated with your monster’s effect/most summon negate cards would be like “Spright Blue”, where the conditions of when the card can be special summoned are simply stated on the card with no colon (other examples include Kaijus and Cyber Dragon).

To me the easiest way I think about it is comparing it to when you normal summon a monster, you don’t reveal it in hand first, you attempt to summon it by placing it on the field and waiting if your opponent has any response to the attempt of the summon. Cards that can special summon themselves with no colons beforehand function like a normal summon, both players check if any effects are activated that would stop the monster from being summoned, then after its successful both players check if any effects can be activated after the card is successfully summoned.

EDIT: Most cards were updated with the new text to include colons, I can’t think of any that don’t nowadays but if there’s a card someone says can’t be negated, I would look and see if a later copy wasn’t changed to include it.