Minimum hamming distance of multiple binary words
You are correct.

Here's what's going on.  There are two subtly different questions here.  The Stack Overflow question is asking: "given a set of bitvectors that form a Hamming code, how do I find the minimum distance?".  In contrast, it sounds like you might be asking "given an arbitrary set of bitvectors, how do I find the minimum distance?".  Those aren't the same question.  Unsurprisingly, they have two different answers.

The answer on Stack Overflow you're referring to isn't 100% clear, but I suspect it is making the following claim: if you have a set of bitvectors that form a Hamming code, then the minimum distance can be obtained by computing the Hamming weight of the xor of all the bitvectors.  I don't know whether that claim is true, but let's suppose it is.  That still doesn't help you with (what I think is) your question, because in your problem you have an arbitrary set of bitvectors -- they might not form a Hamming code.

It is easy to find counterexamples to demonstrate that the claim doesn't hold in general, for an arbitrary set of bitvectors.  For instance, your example is a perfectly good disproof of that.

There are techniques for finding the minimum distance of a set of arbitrary bitvectors (at least, there are heuristics), but hopefully this is enough to answer the question you asked.