I assume your heater has a flow sensor of some sort that tells it to start up when it senses water flow. In theory, your hot water piping is completely isolated from the cold water side. Since your heater tries to fire when you use cold water, I'm guessing there is some kind of crossover between the hot and cold systems that shouldn't be there.
Obviously, I'm not there to actually look at your system, but, thinking logically, I suspect that at one time your old water heater had some kind of recirculating system to provide hot water whenever you turn on the hw tap. My house has this. In Arizona where water is precious, some think this is better than having to wait for the cold water to flush through the pipes before the hot water gets from the tank to the tap. One way or another, your heater is sensing water flow when it shouldn't which tells me that, somehow, the separation between your hot and cold water piping is compromised.
That's my two cents anyway.
Edited to add, "even when no one is using water" do you have an icemaker? A drip irrigation syster on a timer? Anything else that might use water without a concious act by you?
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