I heard that operating refrigerator or water pump can be critical for modified sine wave inverter, what do I have to know about this?

It is worth to mention that these appliances are typically inductive loads incorporating a motor inside which would require true sine wave inverter. Otherwise, the motor in these units would easily overheat, give out a humming noise and eventually would get damaged if operated continuously. You should encounter an issue when powering refrigerator if you are not familiar with this subject. In refrigerator there is a motor which powers compressor, this compressor needs to overcome various compression values when start up, so the current drawn by the motor might get 3-5 times higher than steady value. It is very difficult to calculate how big inverter should you choose using the rated power of refrigerator (e.g. a 300W inverter will not start a 100-150W refrigerator). Generally we recommend you to use a min. 600W or better 1200W inverter for starting a refrigerator in all circumstances. The situation is similar with water pump; it takes high surge current, often lasting several seconds. Pulling water up from great depth would also make pump's working more difficult and it would take higher start-up current. It is not at all impossible that e.g. a 1000W inverter won't start a 500W water pump.