Mathematicians use words differently. What you're talking about is what the article calls a generalized continued fraction. Generalized continued fractions are not unique, and there are actually many of them for pi, which have different rates of convergence. The article here contains a (different) nice generalized continued fraction for pi, and an algorithm for converting from generalized continued fractions to standard ones, thereby computing the exact standard continued fraction for pi as a result.

(These were later edits to the article, so if you read it early on, you may not have read those parts.)