Implicitly defining a number via it's decimal form typically relies on their being a pattern to follow after the ellipsis. You can define a different number with twos in it, but if you put an ellipsis at the end you're implying there's a different pattern to follow for the rest of the decimal expansion, hence your number is not the same number as the one without twos in it.

That's a decimal approximation of Pi with an ellipsis at the end to indicate its an approximation, not a definition. The way the ellipsis is used above is different. It's being used to define a number via the decimal expansion by saying it's an infinite sum of negative powers of 10 defined by the pattern before the ellipsis.

So we have:

0.101001000100001000001 . . . = 10^-1 + 10^-2 + 10^-3 + 10^-4 +10^-5+ . . .

Pi, however, is not defined this way. Pi can be defined as twice the solution of the integral from -1 to 1 of the square root of (1-x^2), a function defining a unit semi-circle.