> L is the smallest number in E such that L is greater than or equal to all numbers in S.

And note that this doesn't imply that L is in S!

So here we have S = { 0, 0.x, 0.xx, 0.xxx, ... }. E is R, the set of reals. The smallest number in R greater than or equal to anything in S is 1. That doesn't mean 1 is in S. If 1 is not in S, then we're taking a notation based on condensing the names of the elements in S, like "0.x...", or 0.x with a bar over the x, and making that notation denote something not in S, namely 1.

This doesn't appear incompatible with the limit-based construction.

Are there instances in which this supremum approach toward continued decimals disagrees with the limit approach, in establishing the value?