Can I put "rechargeable alkaline" batteries in a charger that only says "Ni-Cd/Ni-MH"?

yes there is a fundemental difference. Nicad and NiMH are fully charged at 1.2 volts. An alkaline at 1.2 volts is essentially dead. you would need to buy the rayovac charger to be able to get any benefit from their batteries.