Sunstar Lightsmith’s ability resolves before the spell that caused it to trigger. It resolves even if that spell is countered or otherwise leaves the stack without resolving.

Sunstar Lightsmith’s ability will count any spells you’ve cast this turn, which may include Sunstar Lightsmith itself. It doesn’t matter if the other spells resolved, didn’t resolve, were countered, or are still on the stack. If Sunstar Lightsmith was the first spell you cast this turn, the next spell you cast this turn is your second spell.