Difference between quick lime, slaked lime, lime water and soda lime

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I completely understand your frustration as all those names are trivial names (non-systematic.)

The "lime" is derived from the use of Limestone https://www.etymonline.com/word/lime

I would have to mention that even those names are associated with chemical names that are not pure and contain some impurities.

Lime = quicklime ($\ce{CaO}$ - Calcium oxide) is made from limestone ($\ce{CaCO3}$ - Calcium carbonate) by decarbonisation at high temperatures.

To get slaked lime = hydrated lime ($\ce{Ca(OH)2}$ - Calcium hydroxide) you mix Lime with water, resulting in a white powdery substance.

In another process, you can make lime water - where you add lime into a greater amount of water resulting in water solution/suspension of hydrated lime $\ce{Ca(OH)2}$ in water - this is also called milk of lime.