A power station is a battery and an inverter in one case. If you already have a leisure battery in a van or a boat, a standalone inverter is the cheaper half of that, and usually the better buy.
Pure sine wave, not modified. Modified sine wave is a coarse approximation of mains. Induction hobs and microwaves are sensitive to it in a way phone chargers are not, and the failure is often not a clean refusal but running hot, running oddly, or working until it does not.
Size it to the appliance, not the battery. The continuous rating has to meet or beat what you intend to plug in. A 2,000 W hob needs a 2,000 W inverter whatever the battery behind it holds.
Check the surge rating too. Things draw more at switch-on than they do running, and an inverter that only just covers the running figure can still trip at the start.
The cabling matters more here than with a power station, because it is on you rather than the manufacturer. A 2,000 W inverter on a 12 V battery is pulling well over a hundred amps, which is not a job for thin wire or a long run. If that sentence is unwelcome news, buy the power station instead.