Only - it matters
"If I knew only" is the title of a popular ACES convention session on usage and grammar run by Merrill Perlman, director of copy desks at the New York Times. And, as you can judge from the title, she takes on the bugbear of where to put that four-letter word.
It's a matter of some debate -- 200 years' worth actually. Arguments continue to be made that the word often works more naturally earlier in a sentence than directly before the word it is supposed to modify. And as we become a more spoken-English society rather than written English, American Heritage probably is right that he only had three of those is going to be found just as often as he had only three of those. Eventually, like the over/more than and another/an additional usages (to name just a couple), it will become something copy desks are told not to fret about (some probably already are being told that as the workload just increases after layoffs and buyouts).
But here's an example from an explanation of some high school football rule changes in my local paper that shows how confusion can ensue. The setup -- the high school association is passing a rule that would allow spring football practice with pads and helmets:
Players may only wear helmets for the first three days. Once those three days are completed, players may wear full pads for a maximum of seven practices.
One reading of that first sentence is that they are allowed to wear helmets for only three days -- the first three. (Try saying it with an upturn in your voice on "three days" -- you get a different impression than if you say it with a downturn on those words.)
One way to solve it is to move "For the first three days" to the front: For the first three days, players may only wear helmets. That forces your tenor and cadence into the downturn that makes the meaning clearer. But, of course, that can be difficult to transmit in writing.
So the clearer adjustment -- in both structures -- is to move "only" between "wear" and "helmets." Players may wear only helmets for the first three days. Once those three days are completed, players may wear full pads for a maximum of seven practices.
(Yeah, yeah, I know: "No other clothing?" But this is lesser of two ambiguities for me, and it's less likely to be misread.)
It's a good example to keep in mind when you are struggling with such things.