What are the different uses for the 110 vs 114 vs 116 link bike chains? I used to just leave those to the bike shop guys .. but nowadays .. I would prefer learn these basic skills myself

Currently I was searching for a replacement chain for my bike .. was told that for my MTB .. just get a 114 chain .. but I am really curious about all the different lengths .. so would appreciate if someone can help shed some light on the different use cases for the different lengths
This is one of those topics where the answer has changed with the new directions that bike drivetrains have gone in recent years.

With any chain packaged for consumer use, the basic idea is they come long enough to cover all the common use cases, and you should expect to cut it down to fit your bike. On a derailleur bike, the right length is usually seen as being a function of chainstay length, the size of the largest chainring and cog, and whatever extra length the rear derailleur maker recommends. On suspension bikes, the effect that suspension travel has on effective chainstay length must be accounted for.

Optimal chain length itself is a topic there are many questions here about, and there are more ways of doing it that might lead to different results than might be initially suspected. For example, for road bikes where you want to have wheel/cassette swappability to take full advantage of the rear derailleur's maximum capacity without having to swap chains along with wheels (for example to swap in an 11-28 cassette for a climbier day versus 11-23 for flat), in the eyes of many it makes more sense to size off the small/small combo, even though no manufacturer recommends this approach in their instructions.

However, what's happened lately is there are more bikes with drivetrain setups that feature both "big" chainrings and cogs, sometimes alongside tires with bigger  outside diameters than in years past, and the longer chainstays that come with them. For example, an extreme example is in the shop I work at, we handle a certain "adventure touring"-ish bike that comes with a 48t large ring, 34t large cog, and 700x48 aka 29x1.95 tires with chainstays even longer than strictly necessary for that tire size. It needs a 118 link chain, which exists but is uncommon, and it's a frequent mistake for people and even mechanics to go to install a replacement chain onto it and find the one they picked out comes too short.

From this you can probably infer how different bike categories creep into needing the less common 116 and 118 link chains even though historically 112 and 114 have been enough. Honestly this is an area where a lot of mistakes are still being made and what appears on retail shelves hasn't necessarily caught up to the greater diversity of bikes out there. I don't think it's so much a matter of manufacturers or retailers pinching pennies on the extra links; probably before long there will be a new equilibrium and most chains will come at 116 or perhaps 118.