Motivation for using size_t uint32 uint64 etc
These are for platform-independence.
size_t is, by definition, the type returned by sizeof. It is large enough to represent the largest object on the target system.
Not so many years ago, 32 bits would have been enough for any platform. 64 bits is enough today. But who knows how many bits will be needed 5, 10, or 50 years from now?
By writing your code not to care -- i.e., always use size_t when you mean "size of an object" -- you can write code that will actually compile and run 5, 10, or 50 years from now. Or at least have a fighting chance.
Use the types to say what you mean. If for some reason you require a specific number of bits (probably only when dealing with an externally-defined format), use a size-specific type. If you want something that is "the natural word size of the machine" -- i.e., fast -- use int.
If you are dealing with a programmatic interface like sizeof or strlen, use the data type appropriate for that interface, like size_t.
And never try to assign one type to another unless it is large enough to hold the value by definition.
The motivation to use them is because you can't rely on int, short or long to have any particular size - a mistake made by too many programmers far too many times in the past. If you look not too far back in history, there was a transition from 16 bit to 32 bit processors, which broke lots of code because people had wrongly relied on int being 16 bits. The same mistake was made thereafter when people relied on int to be 32 bits, and still do so even to this day.
Not to mention the terms int, short and long have been truly nuked by language designers who all decide to make them mean something different. A Java programmer reading some C will naively expect long to mean 64 bits. These terms are truly meaningless - they don't specify anything about a type, and I facepalm every time I see a new language released that still uses the terms.
The standard int types were a necessity so you can use the type you want to use. They should've deprecated int, short and long decades ago.