which kinds of electrical equipment need a pure sine wave inverter to work correctly?

Some people claim that certain loads "may" not work as well, or "may" be damaged, with anything other than a pure sine wave.

Since the power coming out of my wall sockets is significantly different from a pure sinewave, I suspect these sincere and well-meaning people are merely repeating propaganda from the manufacturer of a pure sinewave inverter, much like people repeat urban legends.

There is only one kind of device that I know doesn't work as well with a non-sine-wave inverter: devices that use a "capacitive power supply" -- see How efficient is a capacitive power supply? for details.

Since a "capacitive power supply" has a power factor near 0, it is questionable whether any "capacitive power supply" meets EU-mandated power factor laws, such as EN61000-3-2.

I suspect that all products -- as long as they were designed after EU-mandated power factor laws went into effect -- should work just as well on "modified sine wave" as with "pure sine wave" inverters.

Of course, I can't possibly know about every product ever designed -- if there is any specific product (that was designed after those EU-mandated power factor laws went into effect) that can't work or doesn't work as well with "modified sine wave" power, I would be interested. If anyone can explain why it doesn't work, in enough detail that I can try to avoid that kind of failure, I would be fascinated and grateful.

The modified sine wave inverters generally cause more power loss in your products' power supplies. So the inverter itself may not be any more efficient, but the equipment running on a pure sine wave inverter will most likely run more efficiently. This is especially true for inductive loads, such as all the equipment you listed.

I'd guess that most of your equipment with linear AC/DC power supplies will work, but perhaps not as efficiently or with reduced performance and increased heat. Many radios, for example, will sound worse when run off a modified sine wave converter. Other products that use switching supplies may not work. I'd recommend ponying up the extra money for the pure sine wave inverter for anything other than an emergency power supply you need to use temporarily when power is lost, for example. Especially as you are looking to drive inductive loads.

I'm not an expert on inverters, but I know there are plenty that run on 12V designed for the RV community.

Generally speaking, anything with an inductive or motor load, pure sine wave is better. A load that first rectifies the input (PC power supply, phone charger etc) a modified output would be sufficient