I use a double-sided clip. Bills go in one side, cards in the other, although I don’t have a wad of 20s like in the picture. It resides in my front pocket, which is a safer place from pickpockets than in your back pocket. Pictures go on my phone.
For those of you concerned about having back problems due to having a large wallet in one of your back pockets, put the wallet in a front pocket. It has the added advantage of being harder to pick-pocket as well. Plus, KISS! I beleive that this helped to releive my back pain.
IANAD, Nor do I play one in any drama. IHTH, 48.
I went to a chiropractor once. He told me that my wallet was too thick and would lead to back problems. Then he made it a lot thinner for me.
It’s a little archaic, but here in the U.K., a wallet can be really large, capable of holding unfolded A4 or foolscap sheets. The differences between such a wallet and a case are that the wallet is soft, not rigid, and has neither handle nor strap.
I’ve always called this a wallet, and thought a billfold was just a more minimalist type of wallet, for just your ID, some currency, and maybe a bank card or two.
You can also check out the thin wallets made by All-Ett, usually for sale on other websites like Amazon, eBay, etc. I just got mine a week or so ago, and, with all my cards and a few bills, has to be well under a quarter inch in thickness. It feels as if it’s nothing more than a couple of pieces of cardboard in my back pocket. With the same contents my old wallet was at least twice as thick. The only minor drawback is that it’s black nylon instead of leather, so if you like a “dressier” looking wallet it might not be for you. I doubt if you can make a truly thin wallet in leather that will hold up to daily being used and sat upon.
For me, its even more minimalist. 
A billfold (to me), is just for holding paper currency, and nothing else.
Something that held more than that would be a wallet.