Why is it called "christening" instead of "baptism" when it is for a baby?
Solution 1:
This is all interesting, but your choice of what terminology to put on the baby's card/poster has literally nothing to do with dictionary definitions or linguistic etymology of these terms.
You should use whatever term the parents and/or their church prefer, full stop.
Why?
Because otherwise you're digging into a minefield of non-linguistic arguments that I guarantee you don't want to get into. Churches may prefer one term over another for inscrutable theological reasons, or for reasons buried in historical conflicts between denominations, or for no good reason at all.
Here's an example: my (former) church has a "baby dedication" ceremony for infants, and would strongly object to either a "christening" or an "infant baptism", since in their belief baptism is reserved for someone old enough to decide for himself or herself. Other churches are sure to have very different preferences, equally strongly held.
Solution 2:
By rights this ought to be a comment, not an answer, but it's likely to run too long...
I'm surprised that the Etymonline entry for "christen" doesn't mention the root of the word "Christ" itself: it means "anointed". The sprinkling of water that's done at a christening resembles an anointing much more than it does an immersion baptism. In fact, the baby is generally also anointed with chrism.
There's also the term "Christian name", which is no longer in common usage in US English, but is interchangeable with "first name" or "forename" (as opposed to "family name", "last name", or "surname".) It's an interesting chicken-and-egg question: is it your "Christian name" because you were given it at your christening, or were you "christened" because that's when you were given your Christian name?