A monster is considered "successfully" summoned if the summon was not negated or did not "fizzle". If the effect is negated and can only be activated while the monster is on the field, it is still considered successfully summoned, because the summon itself was not what was negated.

If the summon fizzles (like if you play Monster Reborn, but your opponent uses Ojama Trio to fill up your Monster Card Zones), it was never on the field. If the summon is negated (e.g. Horn of Heaven), it counts as never being on the field. A monster that "can only be Special Summoned by" cannot be summoned from the Graveyard if it was not "successfully" summoned, because it was never on the field.

By the time a monster is successfully summoned, a monster's continuous effect has already "take effect". This is why you cannot use Trap Hole on Jinzo. But you can use Horn of Heaven, because technically Jinzo is not on the field yet for the continuous effect to "take effect".

This basically applies to any card use in the game, successful = not negated, not fizzled.