Differences between rice/pressure/multi/slow cooker?
There are several different categories of these devices, but they do all tend to look quite similar. Prices can vary by a very significant amount, though. So.

Rice Cooker

The simplest kind of rice cooker is the kind with just a switch on the front to turn it on. Mine is like this - when plugged in, it's in "warm" mode, press the switch and it goes to "cook" mode. Rice cookers are designed to boil their contents, then wait until the water's all absorbed/evaporated and switch back to a keep-warm mode. That's all you need in order to cook rice, so it's all a basic rice cooker will do. You can cook other things with it, if you bear this in mind.

Rice Cooker with other functions

Depends what the other functions are. I've most commonly seen rice cookers which can also be slow cookers, in which case they heat more gently and disable the automatic shutoff used when cooking rice.

Slow Cooker

A slow cooker, commonly also called a crockpot after a popular brand, is a device that looks quite a lot like a rice cooker but usually has a fairly chunky ceramic cooking pot instead of a thin metal one. Slow cookers are designed to heat up to a moderate temperature and stay there for long periods of time, allowing you to gently cook something like a stew or a shoulder of pork for hours upon hours without having to use the oven or dance attendance on a saucepan. Slow cookers often have two or more temperature settings - mine has "high", "low" and "warm", but they can come with more. A common extra feature is an even lower temperature setting designed for making yoghurt.

As mentioned above, a common combination is a rice cooker which can also be a slow cooker.

Pressure Cooker

A pressure cooker is a very different beast. Pressure cookers have a sealed chamber (typically with a big, heavy lid which you put on and twist to lock it into position) and a pressure valve. They heat up and form a high pressure environment, which raises the boiling point of water. Because the water stays liquid at higher temperatures inside the pressure cooker, food cooks a lot faster. A lot faster. This also has impacts on flavour and texture of food, sometimes positive, sometimes negative.

To answer one specific point in your question, you can cook rice in all pressure cookers, in much the same way as you can cook rice in all saucepans and all microwaves, regardless of if they have a "rice" button.

You can get stovetop pressure cookers, which require careful control of the heat from the burner to maintain the right pressure inside. People tend to be wary of these as we worry about them exploding, although modern cookers have excellent safety valves which will vent long before there's a risk to the integrity of the pot.

The easier kind to use is the electric pressure cooker, which heats itself up and takes care of maintaining pressure automatically. These come in a variety of models with more or less sophisticated features including multiple pressure levels, automatic venting/keep warm/cooling at the end of the cooking time, and various other cooking functions, which leads us to...

Multicooker

Any one of a number of devices which aims to combine one or more of the above categories, plus more, into one device. A rice cooker with a slow cooker mode is technically a multicooker. Some multicookers claim to do techniques usually done on the hob and be able to roast things and deep fry things. Whether this is successful or not is something you should really check the reviews about.

The popular multicookers these days are usually electric pressure cookers with more modes. They may have a hotter heat setting to use for frying and searing with the lid off, or a slow cooker mode which lets you put the lid on but won't build up pressure as it would in pressure cooking mode. Some have a rice cooking function, although typically they will pressure cook the rice rather than simply boil it as a standalone rice cooker does.

How to choose one

First, decide what you want.

Then, decide how much you can afford.

Then, read the feature lists. If a rice cooker can be used as a slow cooker it will tell you, because this is more expensive so they need to advertise that capability. Electric pressure cookers seldom come without some claims of being at least 6-in-1 devices. Mine's one of the most basic on the market and claims 6-in-1, in reality it's only capable of sauteing things (with the lid off) or pressure cooking them (with the lid on). So check carefully. Read reviews. There's no substitute for some research work in choosing a device for your kitchen.

You may wish to have multiple objects so you can use them all simultaneously - which means you can gradually acquire a very affordable slow cooker, rice cooker, pressure cooker, whatever, one by one. But that takes up lots of space as well, which maybe you don't have, so you might want to spend more money and get something which has many modes for different ways of cooking.