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Well done! I think you can still use the term "air conditioner", but you can phrase it like this:

"I used the heating function of my air conditioner."

The reason some native English speakers use the term only for cooling, is because air conditioners in some countries don't have heating functions. In some countries, they use other heating devices, such as heaters and radiators. In my opinion, it's okay to use the term for both heating and cooling. You just have to be specific about the heating function when talking about using it for heating.

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Hmm, maybe use “heater”? But if ACs really have heating functions in Japan, then it’s not wrong to use “Air Conditioner”.

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The internet tells me that "climate control system" is the generic term for the appliance. I am not an expert, but I think it's accurate to say that some of these are separate units that can be installed in a window or on a wall. Others are built into the structure of the building. We have our own confusing terminology: the fairly new term "heat pump" refers to a climate control system that heats and cools.

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In the U.S., these machines are called “heat pumps.”
See the first paragraph here:
https://en.wikipedia.org/wiki/Air_conditioning
“Heat pumps are similar in many ways to air conditioners but use a reversing valve, allowing them to both heat and cool an enclosed space.”

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This one is a bit tricky, I think it should still be called the air conditioner since "conditioner" is a temperature neutral term. In general though, people are used to call it the heater when it is for heating. To give you some background, in western countries like Canada, if it is a place cold enough that require heating, the house usually comes with centralized heater. However, you actually have to install your own cooling system separately. For places that does not require heating, the air conditioner that they buy will only be used for cooling purpose. So it will take some time for people to get used to it having dual functionality.

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We usually just call it a "heater", or you could also say it's "an air conditioner with a heater", or a "split A/C and heater". At least in America, they're usually separate machines, though that has been changing recently. If it's buildings (like the ceiling), we call it "central heating and cooling".