From your own definition, I have a convenient way of storing wealth that is my money that is being used to pay my taxes.
If I did not transfer that to what you call 'actual wealth', then did my wealth go up or down after I paid my taxes?
Also, when I purchase a stock that goes down, did my 'actual wealth' stay the same or go down as well?
Obviously, when you are taxed, you lose wealth (or at least a potential claim on future wealth if you pay in cash). But that's different from the statement that, "Everytime you purchase something you give up wealth". If one properly understands trade, and assuming the trade is voluntary, the proper statement would be, "Every time you purchase something you gain wealth" since if you weren't better off from the trade you wouldn't make it in the first place.
If you purchase a stock that goes down, your actual wealth remains the same, just as it would if the stock went up. Your potential wealth has changed, or at least it is likely to once you sell.