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I'm actually annoyed by this. Since the next digit is 6, he should be rounding to 3.141593, not truncating at 3.141592.
The difference between the two is 0.000001, and the difference in quality of approximation is roughly 0.0000003. So I don't think it matters as a practical preference. Aesthetically, the string of digits of Pi has a cultural significance of its own, quite apart from its numerical value, so I think it's preferable to memorize a number that's actually a prefix of that string.
Also, only if you truncate can you truly say that you've memorized "the first six digits of Pi after the decimal" as opposed to "An approximation of Pi to plus or minus 0.0000005 tolerance".
The sixth decimal digit of pi after the decimal point is 2.
The value of the millionths place in the decimal expression of pi when rounded to the millionths place is 3.
The reason I know pi out to the sixth decimal digit past the decimal point is because the digits to the fifth fit into an easy rhyme, but the fifth decimal digit after the decimal point is a 9. If it were to round up there would be a little cascade, so it's good to know the next digit.
If Harry knew to round up the millionths place in the decimal expression of pi when rounded to the millionths place, he would know pi out to the ten millionths place, or the seventh decimal digit following the decimal point.
Rounding errors at the millionths place are errors of less than one part in a million.
The original text remains correct.
One doesn't need to remember what the digit past the rounded one is in order to memorize the rounded figure.
Did I say that the text was incorrect? I'm 99.999% sure I didn't, but looking back . . . no, still looks like I didn't. All I said was that I was annoyed, and what I think Harry should have memorized instead.
If this annoys you and so you always stop your approximations at a point where the next digit is 0..4, I think this biases your estimates of numbers (in a way that doesn't really matter for most purposes).
Why would I do that rather than simply round correctly for the number of sig figs I'm dealing with?
I don't particularly care how many digits Harry want to go to, just think he should pick 3, or 3.1, or 3.14, or 3.142, or 3.1416, or 3.14159, or 3.141593, or 3.1415927, or 3.14159265 . . . etc.
Obviously Harry should go up to the Feynman point.
It's a risk for a hypothetical person who is bothered by having to round, which isn't precisely the thing you're bothered by. A person who doesn't decide in advance how many digits to use/remember.