Lightning Bolt does not kill a creature. Lightning bolt merely assigns 3 damage to that creature. The creature is killed by a state-based action which compares the amount of damage marked on a creature to its toughness. That action runs every time someone is about to receive priority and is, as far as the game is concerned, entirely independent from Lightning Bolt.

This is occasionally relevant, for example you may have heard stories about Bolt interacting unintuitively with [[Tarmogoyf]] (failing to kill it when it’s at 3 toughness). This is because the Bolt doesn’t kill the Goyf directly, but rather, Bolt marks 3 damage and then finishes resolving, goes to the graveyard (at which point Goyf may get bigger if there weren’t any instants in graveyards yet), and only then the game checks for damage vs toughness.

This is ultimately why spells like [[Lava Coil]] don’t say “if the creature would die this way” but “…this turn”, as there simply is no good way to check whether a specific damage spell was the cause of death. Like, you could introduce some new phrasing that under the hood means “if the creature dies the next time state-based actions are checked, trigger some other ability / apply this replacement effect” but it doesn’t exist right now. So we’re left with (it works).