Such specs are extremely up in the air, as SpaceX is constantly iterating their designs through extensive testing in Boca Chica. They've been pushing raptor 2 chamber pressure, adding more engines, lengthening the craft, etc.
So it is extremely hard to say what the specs of Starship will be when it is fully operational, around a year to three years or so, pending successful testing, let alone in 2060.
I would be apt to point out that you seem to have outdated or inaccurate information - ITS was only a very brief name of the system, later renamed to BFR, and then renamed to Starship, which seems to be the final name, as it has been stuck to for some years now.
For very technical estimates on the craft, I'd recommend looking at resources from Everyday Astronaut, NASASpaceflight, and others, like a personal favorite of mine, Apogee on Youtube.
The best we can figure, SpaceX seem to want to keep Starship, or at least the basic idea of it, their main workhorse going forward for decades. Elon has talked about vaguely making it bigger, and that there will be some eventual engine that they will employ to make life multiplanetary, but it is not clear if that engine will be just an upgraded raptor, a whole new engine model, something like a nuclear engine, or something not even used on Starship, something like a nuclear engine on a Mars transit cycler built by Starships.
The only clear answer I can give with some degree of confidence - if Starship succeeds, and is not superseded by a very different fully reusable architecture, it will likely be similar, but much more refined.
I'd imagine that the system would look much different, but still use the basic idea of a two stage fully reused vehicle, that refuels in orbit. We might see things like the removal of parts consider redundant, like the wings if SpaceX finds that they can get by without them somehow.
All in all, it's rather up in the air. Much can change, SpaceX could get complacent in the 2050s and get outdone by a new company - a lot of things can change on that timeframe, especially when there are so many unknowns.
You mention at the end of your question "what will make this the killer rocket of the 21st century?"
To me, this speaks to reliability. Soyuz is one of the most reliable rockets out there. It has flown hundreds of flights without fail. Though if you look at the thing, you find that the cosmonauts that ride in the thing are tightly packed, and have to use a stick to manipulate the controls. It is outdated Soviet technology - but it is tried and true - it works well enough.
I can see Starship becoming something like this - they are trying their darndest to make the rocket as simple as they can, removing all unnecessary weight and parts. Their philosophy is "the best part is no part."
If you think in terms of murphy's law, then this is a rather good strategy.
So I'd say that in trying to envision what a refined version of Starship would be, I'd say bigger maybe, and as simple as possible. Maybe they might reach a point this decade where they stop pushing the tech, and focus on making it as hum drum as possible, lowering the cost and making it as stalwart of a machine as they can.