Both statements are correct. The best way to understand how these two statements can coexist is to understand the concept of a gas pressure.
Now to understand pressure we look at a container full of gas molecules. Gas molecules do not behave at all like solids or liquids. In a gas the molecules are not attracted to one another so they fly around at extreme speeds bouncing into objects and other gas molecules. These collisions are elastic so no energy is lost during collisions.
Every time a collision occurs some kind of energy transfer takes place between molecules. However, on a macroscopic level, there are so many collision taking place that they average out to zero energy transferred. Imagine that a gas molecule is about to hit the wall off the above container. We know that when the molecule hits it will bounce off and head in the other direction just like a bouncy ball. The wall will also feel a force due to
Newton's second law. However on the other side of the container the exact same thing is happening. In fact the same thing is happening on the outside of the container as well. All these collisions exert a force but they all cancel each other out.
Now let us apply this to your first deffinition. As you stated air pressure is caused by the weight of the air molecules above. Gas molecules are attracted by gravity towards the earths surface. As a gas molecule is pulled toward the surface of the earth chances are it will hit another gas molecule and bounce off of it in another direction. Now lets say that in this particular collision the first molecule hits the top of the second molecule. This causes the second molecule to travel downwards even faster than the first molecule. This happens again and again until the molecule bounces of the surface of the earth. This is how your first definition is derived. The key is to remember that this is a gas pressure and thus is from all sides.
This is the hardest concept to grasp because when someone hears that there are hundreds of pounds of air above them they imagine hundreds of pounds of steel plates on their shoulders. Do not think of it like that. If a bouncy ball falls on your head it pushes you down. However if it misses, hits the floor, bounces up and hits you the two forces cancel each other out. The trick is to realize that so many collisions are occurring at such a minute scale that you do not "feel" the pressure of the atmosphere.
Solid objects are very good at resisting an even force from all directions. Have you ever heard that you can't crush an egg if you squeeze it from all directions? The same concept applies to to your body. The atmosphere is pushing very hard from all directions (even from inside your lungs!) but they all cancel out.
To contrast this imagine a steel drum with just a few gas molecules in it what would happen?
Now despite this being cool notice that the sides of the barrel collapse as well. This means that air molecules were pushing from the side but there was nothing to push back from the inside. We can see from the imploding barrel that the atmosphere is compressing us with enough force to crumple a steel drum. However because this pressure is exerted from every direction the forces are cancel out and we don't feel a thing.