I think you’re being a little too upright.
After all, it’s common language to speak of soldier’s “baptism by fire”, aka their first real combat experience. Baptism by fire is also a common term used to denote any kind of first ordeal.
Investipedia even lists it as a term. I’ve never heard anyone complain about the use of that term.
Similarly, Baptism in the Holy Spirit is your first real conscience experience of the Holy Spirit working through you and in you. Nobody’s ever claimed it as a replacement for or a supplement to your first baptism in water.
I think the use of the phrase baptism is more used in these 2 cases because it takes the idea from sacramental baptism that your life is really changed after (your life before and your life after are almost like 2 different books). A soldier is forever changed after their first real experience in combat, and so is a person after baptism in the Holy Spirit. That’s all it’s really referring to.