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What all of the above have in common is managing the use of the power efficient, but low performance integrated GPU and the high performance, but power hungry dedicated GPU in a laptop. The goal is to have the both benefits and be able to switch between as needed. The difference is in how they achieve this, and drawbacks there are to them.

Optimus is an Nvidia proprietary technology, and therefore limited to Nvidia graphic cards. Can be found in most laptops, including those with AMD CPU, as long as the laptop has Nvidia GPU.

Benefits

Allows switching between iGPU and dGPU in real time, which means no restart is required. The laptop will decide what GPU should be used for each application. Sometimes it doesn't work properly, but that can be easily rectified. Nvidia control panel application allow manually setting which GPU to use. Such fix is easy and the control panel remembers the settings, so needs to do it only once.

Drawbacks

Besides sometimes picking incorrect GPU, which can be manually fixed, there is a performance hit. Due to how optimus works, all the display output is routed through the iGPU. dGPU processes the image, and then pushes it to the display through the iGPU. This introduces performance penalty, which can be as high as 10%. The penalty can sometimes be bypasses. Some laptops, like those from Lenovo Legion lineup, have at least one or more laptop ports routed directly to the iGPU. By connecting an external monitor to one of such ports, this penalty can be bypassed.
iGPU by its nature doesn't have its own video memory. As a replacement, iGPU uses the system RAM. This can be observed as “reserved ram,” which means a portion of your memory will be off limits for other uses. Can be an issue in laptops with low memory to begin with, especially for laptops with 8GB RAM or less. When there is not enough memory for Windows, Windows will reserve a portion of the storage drive to work as a stand in for RAM. Not noticeable for general use if laptop has a decent SSD, but very noticeable when using a demanding application or a game. The bigger the laptop RAM, the smaller the issue of iGPU having a portion reserved
The reserved RAM can be anywhere between 0.5 - 2GB. Sometimes the size can be manually set.

As the name suggests, it's an improved optimus that solves the major drawbacks of the regular optimus. Not as widespread. Usually found only in laptops with Intel CPU coupled with Nvidia GPU. Overall, it's the best option for retaining power efficiency, high performance and ease of use.

Benefits

Allows switching between iGPU and dGPU in real time, just as optimus does. However, does not route the output through the iGPU.

Drawbacks

Can make incorrect decision on which GPU to use when.

Hybrid mode is a Lenovo name for the MUX switch, which means they are functionally the same thing. MUX allows manually turning the iGPU off. MUX is not mutually exclusive with Optimus / Advanced optimus. A laptop can have both MUX and Optimus inside it. While MUX is a hardware switch, it is still controlled through the user interface. Either through BIOS, or through some proprietary laptop application, for example, Lenovo Vantage application.

Benefits

MUX solves the issue with reserved RAM or the performance hit by simply allowing the iGPU to be turned off entirely. Often highly sought after by gamers. In practice, allows to remove the performance penalty of optimus. In case of Advanced Optimus, when working correctly, MUX is not needed. However, with MUX having a manual switch that works in a binary state - either on, or off. Due to this, it is still preferred to have the MUX even on laptops that come with Advanced optimus. With MUX, there is no software logic involved in picking the right GPU that can fail.
Having both MUX and Optimus inside a laptop can negate the lack of Advanced optimus, by allowing to have both the high performance mode without the iGPU bottleneck, as well as power efficiency mode by having optional optimus when necessary.