If we have a cold and are not coming in for a day or two, is it natural to say "I 'd like to take a sick day today" in AmE?
There are two issues: The fact that you will not be in, and how you want the time "recorded."
For most of us, the fact that you won't be in is the key point.
I'd send an e-mail saying "Feeling really poorly - going to stay home today" or something like that. The fact that my time will be recorded as "a sick day" or "as sick leave" in the time keeping system is of less importance. We'll figure that part out later. I'm what's called an exempt employee, and I get my full salary whether I work 24 hours in one week or 60 hours in one week. In fact, most of us work from home when we're sick if we're able to, because our work doesn't go away while we're at home.
For others, how you record the time may be more important. They may call and say "Please put me in for a sick day today. I'm feeling terrible." An hourly employee needs to have all their time accounted for (sick, vacation, jury duty, whatever) or they won't get paid for the time not at work.