Do projectiles released by melee weapons deal melee, ranged or magic damage?

They are considered magic damage by the game. The terraria wiki states that the Terra Blade "is a Sword/Magic Weapon that fires sword-shaped projectiles." Also, if you kill a player with the projectile, it gives a death message that does not contain the word "terra blade" (or similar), rather it gives the name of the projectile, though this is not necessarily related

Just tested this by using different headgears with the terra blade. I can confirm that melee damage boosts the terra blade projectiles, not mage damage.