Firstly, the linked battery doesn't have a smart inverter, so still not a UPS. An online UPS will also have a AC-DC converter (rectifier) in addition to the DC

Firstly, the linked battery doesn't have a smart inverter, so still not a UPS. An online UPS will also have a AC-DC converter (rectifier) in addition to the DC-AC converter (inverter). It usually also has fans to cool those since they are running continuously, even when AC power is available. An offline UPS is basically an online UPS with a relay, so that if the AC power is working, all of the UPS circuitry is bypassed. This reduces the usage of the DC electronics and increases the efficiency when power-loss is rare (since the AC->DC->AC path will always dissipate some power). A line-interactive transformer is like an offline UPS, but with voltage conditioning (e.g. via an autotransformer) to handle voltages that are a bit low or a bit high. Consumer UPSs are (almost all?) either offline or line-interactive.