Where exactly does "Outer Space" begin

If I am correct, I do believe that there are four basic layers in our atmosphere

Tropisphere
Stratosphere
Ionisphere
Exodosphere

(If i spelled those correctly)

And there are several smaller layers that lay between each one

If I am correct, I do believe that the top layers of the exodosphere is the exact point where the earths atmosphere ceases and the vacuum of space begins. There are others who argue that this info is incorrect for various reasons. Do you agree/disagree with this? If so…why? why not?

There isn’t any “exact point” at which we can say Earth’s atmosphere ends and space begins.

From NASA.

That might be a workable definition of space, but what are the differences between near space, outer space and deep space?

The X-prize people set the edge of space at 100km. A privately built ship made it to an altitude of 64km just this week.
Defining the edge of the atmosphere is tricky as it expands and contracts with the phases of the solar cycle

“Outer space” is synonymous with the colloquial usage of “space”, i.e., above the atmosphere. Technically, any point with a position is in “space”, hence the distinction.

I’m not familiar with the term “near space”, but I would presume that it’s synonymous with low Earth orbit, that is, an orbit about as low as you can make it without crashing.

And the meaning of “deep space” varies with context. As far as NASA is concerned, it seems to mean “beyond the Moon” or “beyond where the Earth’s gravity is significant”. It can also mean “far enough away from any mass that gravity is negligible”. Of course, “significant” and “negligible” depend on your context, too.