It's up to the receiver of the gift to change their membership level. The gift just adds to the account balance and will not change a membership level; many memberships, outside of gifts, are subscriptions, so the member will be charged at the current membership level when it expires.
If you want to be at a different level, you should be able to go in and downgrade or upgrade to the level you want and the system will use any credit you have to offset the new term cost.
Does it help?
My son is being threatened and cheated by few people. He is made to pay diamond membership as gift to few people. We want to revert this. Chess.com can you please help?