On average, yes, people will spend more with credit than debit, and more with debit than cash. It's all about the immediate and visceral hit and knowledge of your limits.
Ever go to a place like Dave & Busters? Remember how all those places you used to just put quarters in a slot? Now you buy a card and just slide the card in. Even better: the games now cost odd numbers of credits, like 5.2 or 6.7, so you can't even do the math on the fly.
I am smart. I am good at math. And I still blew something like $50 on games without even realizing it. Because they want you to get caught up in the excitement of the play and not realize how fast your balance is going down. Imagine having to go to the change machine for $50 worth of quarters!
I have never been a fan of cash, just because I'm usually too lazy to go to the ATM. But you know what I used to do? I used to write down everything I spent in a notebook the minute I spent it, and then every night go enter that into a ledger book that I used to track my spending. It wasn't as immediate as paying cash, no, but I knew every single day exactly how much I had left in my budget in each account and how much my purchases affected that. It made it real and visceral, in pretty-darn-close-to-real-time.
If you are trying to minimize your spending, you want to make it harder to spend than to not spend -- you need the difficulty or pain of spending to outweigh the little happy rush you get from buying whatever-it-is. Cash works because if you don't have it in your pocket, you literally cannot spend it unless you leave, go find a bank/ATM, get more, and come back. That "stickiness" -- that built-in pause and the obligation to go to another place and come back -- is often enough to make you think, gee, do I really need it that bad?* Barring that kind of physical barrier, tracking your spending religiously -- as in daily, if not immediately -- works because it forces you to be accountable, and to face the consequence of your spending choice right away. Now that I don't have my little notebook anymore, I find myself much more surprised by the CC bill every month, because I'm just not paying attention at the same level.
tl;dr: figure out what works for you. But don't bullshit yourself into thinking you're immune to all those clever marketing tricks just because you're smart and savvy and paying attention.
*Which suggests that if I were really trying to cut my budget, cash would be the way to go, because it would use my own sloth as a barrier between me and temptation.