Is There a Difference Between Secret Santa and White Elephant?

In the past, our friend groove has tried tdo secret santa, which was like kind of successful. But then there were always like two or three people who were like, oha, never got my gift. That way, whoever shows up will always get a gift. Theyre's like some rules with stealing gifts. You can steal a gift up to like, two imes within the circle, or you can just let go open a new gift. A little complicated. But anyway, either way, bottom line, everyone who shows up will get a gift."