You are correct in that the airdrop and faucet extensions do not work the same.
This will be remedied in the big upcoming release of wallets6, where the airdrop extension and the faucet extension will both work the same way: They will increase the user balances arbitrarily at your request, and it’s up to you to make the hot wallet balance match so that it can back the user balances. It will also be possible to manually debit/credit users from the admin interface by simply inserting off-chain transactions.
Right now, with the software you’re using, the faucet transfers funds from a designated account to the faucet claimers.
The airdrop extension adds to user balances amounts at your request, but for the users to be able to actually withdraw these funds, the funds must exist in your hot wallet balance.
There is no need for frustration. Simply study the extension descriptions on this page, ask me any questions, and test everything on your staging or integration server before using the extensions in production.
So, please let me know if you have any more questions about how all of this works. If you have questions unrelated to this one, please open a new thread.