It's true that collecting ten Ice and Snow Badges makes one the City Lord, but there's a prerequisite. You must first be the Floor Master of the Ninth Layer. Only then, upon collecting the tenth badge, can you directly become the Floor Master of the Tenth Layer—the true City Lord! Whether you're called City Lord or Tower Master is irrelevant at that point; the Ice and Snow Tower will belong to you.
The reason for this rule is that the City Lord must be the strongest member of the Demon Race! Anyone capable of becoming a Floor Master is bound to have considerable ability, which is why they qualify to become the City Lord. If the City Lord lacked great talent and strength, they would be unworthy of the title.
