Is heating water endothermic?
So far I feel like it is endothermic because heat had to be added to
  the water in order for its temperature to go up, so the water
  essentially absorbed heat, but I'm really not sure.


This is mostly semantics. Such exam questions are annoying when they make you argue about words. The term endothermic and exothermic are reserved for reactions by IUPAC. An endothermic reaction is one where the overall enthalpy change is positive. However one can extend it to endothermic process, which is any process that acquires energy (from external surroundings). Atkin's physical chemistry clarifies it nicely "it must never be forgotten that heat is a process (transfer of energy as a result of temperature difference), not an entity"