Assuming it works similar to Strike, what they charge for converting between dollars and BTC isn't dependent on whether it's for purpose of cold storage or lightning.

NOTE, however, that Strike to Strike transactions don't necessarily require any conversion since they can just adjust balances within their system accordingly. I'm just guessing, but I strongly suspect the video presented above - since it shows "Fees Free" - is CashApp to CashApp; or maybe CashApp to Square (since both are Block, inc).

https://strike.me/faq/what-fees-and-rates-apply-to-cash-transactions/

"Certain transactions within the Strike app involve a conversion between cash and bitcoin, such as buying and selling bitcoin, paying Lightning invoices from your cash balance, or making an on-chain transaction from your cash balance. The conversion rate is based on exchange rates offered by Strike’s third-party service providers at the time the transaction is executed and, for some services, a margin applied by Strike, from which Strike and its service providers earn revenue."