I think some concepts here are being jumbled.
A finance contingency gives you a way to cancel your contract and recover your deposit.
Waiving financing contingencies is typically (at least in my market) deemed a "cash offer" - but you can close it using cash, HML, conventional, bags of pennies, however you want.
When it comes to showing proof of funds, you can show a bank account, or a pre-approval letter from various places.
If you waive your contingencies, and can't close, the seller is going to keep your deposit.
In most cases when buying foreclosures, if they see a pre-approval from a HML, you're going to need to re-write the contract as financed. In these cases, it's better to just say you're using commercial financing when you make your initial offer. For most institutional sellers, they're not going to have a big problem with HML vs. Cash, and will consider your offer the same.