A quibble of terminology: fields are rings too, so what you want to say is that it's not a field (or that it's 'only a ring') because the polynomial is reducible, rather than that it's a ring because the polynomial is reducible. It's a ring because it satisfies all the necessary properties--whether the polynomial is reducible or not has no bearing at all on whether the object under discussion is a ring.
As to why it looks similar to the visualization in the article, your hamming distance picture is essentially the multiplication table for a galois ring GF(2^n), where n is width of the bits of the binary numbers you were using.
Note that I say galois ring, not field; assuming that an n (for n even) bit unsigned binary number overflows as such: 0x00..01 + 0xFF..FF = 0x00..00, then your chart is the multiplication table of the ring formed by taking Z_2 modulo the ideal generated by the polynomial <x^n-1 + x^n-2 + ... + x^3 + x^2 + x + 1>, which is a ring since this polynomial is reducible (it has a root of 1).