Whether "Sunday" means "last Sunday" is a moving target. On Monday saying "Sunday, I ..." it is usually clear that you mean last Sunday. Later in the week, it can start to become more confusing.
Likewise, "this Sunday" and "next Sunday" can be the same, or not. On Saturday or Friday, saying "next Sunday" would probably mean more than a week away, not a day or two.
But earlier in the week, this and next Sunday usually mean the same.
The "on" of "on Saturday" is not strictly necessary, but it sounds better to me that way.
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