You won’t find the word “christening” in any official Church document because it’s not a word used by the Church. The word used is “Baptism” because that word describes the Sacrament which is conferred.
The word christening itself came about because some non-catholics deny the efficacy of infant baptisms, and instead do a sort of “naming” ceremony for newborns.
The baptism ceremony should have its emphasis on the Sacrament of Baptism. The emphasis should not be on the “naming” part, although that is an important part of the overall ritual.
In some contexts, the word has found a way into “everyday” Catholic vocabulary. There’s nothing wrong with using the word–the important thing to do is to baptize the child.
In our vocabulary (secular as well as religious) we often use “unofficial” words, and since everyone knows what we’re talking about most of the time, it doesn’t mean very much (“that’s not a shovel, it’s a spade” or “that’s not a spatula, it’s a turner”). If someone says “Father will wear a purple robe on Ash Wednesday” we don’t need to say “no, Father will wear a violet chasuble”. If it’s an RCIA class and we’re going over the names of vestments that might be a time to correct the speaker, but in everyday use, we simply shrug it off because we understand what’s being said.
Sometimes using the wrong word can indeed cause confusion, and in those cases, we should not use a “substitute” word even if everyone does know what we mean.
Baptism is the corrrect word, and it’s the word we should be using. But to casually use the word christening instead usually doesn’t amount to much. In a CCD or RCIA class, I would insist on using the word “baptism”, but in casual conversation, “oh well…”