The Saint Nicholas Center explains, Santa Claus is another name for St. Nicholas, who was bishop of Myra. Over the years Saint Nicholas became "Sant Ni Klaus" which became "Santa Claus". He is the patron saint and protector of children, and known for his generosity and orthodoxy. And yes, he did punch a heretic at the Council of Nicaea. And anyone who says he doesn't exist is sadly mistaken.
This is what I have always taught and continue to teach my children. But realize there are three Santas we talk about.
First, there is the Santa I just mentioned, St. Nicholas, the "real" Santa. Of course he exists. His relics are on display in Italy. And yes, he's real. As real as all the saints with him in heaven. I suppose Protestants would have a problem with him being alive in heaven, but even they must admit he exists, even if "sleeping." But of course, as a Catholic I recognize that God is the God of the living, not the dead (Matthew 22:29-32).
Then there are the guys in the mall who have fake beards and say "Ho ho ho." Nobody believes they are really Santa Claus, they are just actors playing a twisted notion of Santa. In fact, as I explained to my kids from an early age, the "Coca Cola" Santa is not the real Santa. But if you want to call them Santa, yes, they exist. the proof can be found in any mall across the country.
Laslty there's the Santa that delivers presents in the middle of the night on Christmas Eve. Of course he exists. The proof is that there will be presents under the tree on Christmas morning. "But isn't Santa really your parents?" my youngest son asked me. Well, if so, then of course your parents exist, so that Santa Claus exists too.
So by any definition of "Santa Claus" he exists. As I see it, it is misinformation (at best) to tell your children Santa is a myth.