The arrow saying outside is correct, but the other arrow is incorrect. Both arrows point to the outside of the vase. Even if you were to drill a hole in the base of the vase, you would not reach the inside of it. All you have done is create more outside surface area. You can see the inside of the vase if it is made of glass.
Any area enclosed by the surface area is internal.
Even if you put a lid on the vase, it will not make the empty space surrounded by vase, the inside of the vase. Luckily this is a truth that can be easily explained. However I have asked my 16 year old daughter to read so far and she said that she didn't really understand why the curved arrow was not pointing to the inside. If a person can understand this truth, then they can understand that they may be ignorant of some things that are not exactly clear. They may be able to see the error of their thinking.