Only read the OP, so might be retreading ground here...
MS introduced built-in hard drives and good online console play with the OG Xbox, and then added achievements and refined the online aspect with 360, as well as popularizing indies. Perhaps the modern design of triggers on controllers were also theirs last gen, could be wrong though. The 360 controller becoming a default for PC gaming is something of a bullet point, but thats not exactly innovation I guess.
...and that's really about as far as I can go with crediting MS. They had to innovate somewhere to be where they are in the market today, and last gen was where it happened.







