After all this, I am still yet to understand the concept of open/closed wallet, software/hardware wallet, online/offline wallet. I keep getting them mixed up whenever I try to have a grasp of them.
Open and close source as you have been told already means if their source code is made available for the public it's open source while if the source code is not available, it's closee source.
Open Source Hardware Wallets
https://bitcointalk.org/index.php?topic=5288971.0Online wallet are the keast you could use or get as a recommendation for use because they are not safe for your advantage, you will not have access to the private keys because they are connected to the internet and mostly are mobile wallets, some some exchanges also have this kinds and any wallet that will not give you access to your private keys is a custodial wallet, no. Custodial wallets are best, such as you get with cold storage wallet like hardware wallets, you can also use a decentralized wallet like electrum or run of bitcoincore software for privacy.