A simple module is a semisimple module .
A module $M$ is called semisimple if every submodule is a direct summand of $M$
Since a simple module has $\{0\}$ and $M$ as its submodules so it is semisimple.
But why is a simple ring not semisimple because a
A ring is semisimple if it is semisimple as a module over itself
How to construct such a ring?When will the result hold?
In different categories (such as the category of rings or the category of modules) different substructures are important. Ideals are the natural substructures of rings, and submodules are the natural substructures of modules. These substructures are special because they are suitable for forming quotient objects in their categories (that is why I am not mentioning ‘subrings,’ for example.)
“Simple” has come to mean “having no nontrivial substructure,” (or as suggested in the comments “admitting no nontrivial quotient”) so in the case of rings this means it has no nontrivial ideals.
If you want to talk about right and left ideals, then you must realize that we are switching gears and thinking of the ring as a module over itself and no longer like a ring. Notice how the definition you gave for semisimplicity of a ring is entirely in terms of its semisimplicity as a module.
Anyhow, this gives you the answer to your question. The definition of a “simple ring” refers only to ring ideals, and that is not directly connected to its identity as a module.
How to construct such a ring?
The Artin-Wedderburn theorem completely classifies all semisimple rings. They are exactly finite direct products of matrix rings over division rings. This gives you every possible example.
You can make simple rings out of any ring by taking the quotient by a maximal ideal. To get an example where the result isn’t semisimple, you’ll just have to be careful to make sure the quotient isn’t Artinian. One such example is to take the ring of linear transformations of an infinite dimensional vector space and take the quotient by a maximal ideal. This is a non-semisimple simple ring.
There is even a simple ring without nonzero zero divisors. One such example is that of Weyl algebras. Semisimple rings usually have nonzero zero divisors: the only semisimple rings without nonzero zero divisors are division rings.
When will the result hold?
A simple ring will be semisimple iff any if the following conditions hold:
it has a minimal left ideal (or minimal right ideal)
It is right artinian (or left artinian)