A Wizard spell is one with the creature type Wizard. Spells that are Wizard-themed (such as Relic Amulet) aren’t Wizard spells.

An ability that triggers when a player casts a spell resolves before the spell that caused it to trigger, but after targets have been chosen for that spell (if it has any targets). The ability resolves even if that spell is countered.

You draw a card and discard a card while Windrider Wizard’s last ability is resolving. No player may take actions between those events, and nothing can happen.