Whether the tax is imposed on the buyer or the seller doesn't make any practical difference. It's just an accounting detail. One might even say, it's just an accounting trick.
I haven't read this particular book but I imagine what he was driving at with the "bowl" example was this:
Suppose that instead of filing tax forms, etc, the government collected taxes on the spot: There is a tax collector standing beside the counter with a bowl to collect the tax money. The price not including the tax is \$10. The government charges a \$1 tax.
So, scenario 1: The government declares that the buyer must pay the tax. So when you buy the product, you hand the seller \$10, and then you drop \$1 into the tax man's bowl.
Scenario 2: The government declares that the seller must pay the tax. So the seller increases the price to \$11. When you buy the product, you hand the seller \$11. He then drops \$1 into the tax man's bowl, and keeps the remaining \$10.
Either way, the buyer has \$11 less cash, the seller has \$10 more cash, and the tax man has \$1 more cash. Whether the buyer hands the tax money directly to the tax man, or the buyer hands the money to the seller and the seller hands it to the tax man, makes very little practical difference.
We could discuss what portion of the tax is really born by the seller and what portion by the buyer. If today the tax is zero and tomorrow the government begins imposing a \$1 tax, it doesn't necessarily follow that the seller can continue to collect \$10 and the buyer will pay \$11. Increasing the cost to the buyer to \$11 would mean that fewer units are sold -- some number of people will not buy or will not buy as many for \$11. So the seller may conclude that to maintain sales sufficient to maximize his profits, he must drop the price to, say, \$9.60, so with tax the total price is now \$10.60, and the seller is paying 40 cents of the tax while the buyer is paying 60 cents. Or he may have to drop the price further, or not drop it as much. But whatever the optimum price ends up being, whether the government nominally collects the \$1 from the buyer or from the seller has nothing to do with how the tax is really being split between buyer and seller.
There is the technicality that there may be costs involved in record-keeping or other mechanics of the tax. Like if the seller is required to collect the tax and turn if over to the government, then the seller has additional book-keeping costs. If the buyer is expected to keep records and pay the tax, then the buyer has book-keeping costs. But this isn't the main point.