Dimension (noun): A type of direction where each direction of this type points away from (is perpendicular to, shares no vector component with) every other direction of this type.
Dimensional (adjective): The characteristic of a space or object as having some number of dimensions.
Space: An N dimensional space is one where objects in that space have N dimensions of freedom of motion. (For Example: Air molecules in my room have 3 dimensions of freedom of motion.)
Object: An N dimensional object has extension (occupies space) in N dimensions. (For Example: Mario from the original arcade game has two dimensions of extension and cannot occupy volume in 3D space.
So how is the mathematical (geometry) definition of "dimension" different from the physical one?
First, physical dimensions do not exist, and mathematical dimensions do. A mathematician can define the dimensions of a coordinate system as they please, and then those are the dimensions of that coordinate system. In physical reality, there is no coordinate system and there are no objectively determined dimensions. Rather, objects and spaces have some number of directions of extension and freedom of motion and the physical definition of the term "dimension" gives the adjective "dimensional" descriptive meaning.
In Physical Reality, a dimension is just a direction. It cannot bend. In mathematics a dimension is an axis of a coordinate system subjects to the laws of mathematics. If mathematicians want to bend the coordinate system all that matters is that the math be logical and consistent. Einstein showed us that space in our universe was best described using a coordinate system that could curve. The surface of the Earth is best described by a curved coordinate system (latitude and longitude) too. That is OK. A curved n dimensional coordinate system describes the curved surface of an (n + 1) dimensional object. Any curved coordinate system could be describe with a flat coordinate system with one additional dimension.
Non-spatial dimensions do not exist in Physical Reality. A dimension is simply a direction after all. This means that time is not a dimension in Physical Reality. Contrast this with mathematics which can have any number of dimensions. Anything that can serve as an axis on a graph can be a dimension. Heat is a dimension in math. Hardness, attractiveness, profitability, productivity, personal income, population, and military strength can all be thought of as dimensions.
Consider the case of a cheese-burger in a mathematical model. (The author is, at this moment, hungry.) A cheese-burger has the three spatial dimensions that determine its physical size. Additionally, it has a time coordinate that determines its age. It also has HEAT, DELICIOUSNESS, ATTRACTIVENESS, and FRESHNESS which all determine how desirable that cheese-burger is at a given time. Furthermore, these four dimensions generally decrease with time. Finally, you have the three spatial dimensions again a second time with the location of the cheeseburger relative to my location. This is eleven dimensions of cheese-burger. Is a cheese-burger an eleven dimensional object? Yes, in a mathematical model because why not?
However, in Physical Reality the cheese-burger extends in three different directions and thus is three dimensional.
Time is a non-spatial dimension and therefore cannot be a dimension in Physical Reality. What then is the fourth dimension?
The fourth dimension can only be a direction away from (perpendicular to) three other dimensions. We are three dimensional beings in a three dimensional universe.
The fourth dimension is the direction that points away from the universe.
According to Einstein and the last century of Theoretical Physics, there is a direction that points away from the universe in Physical Reality. If you were to pass through a wormhole, you would move in the direction perpendicular to the three dimensions of our universe.
To be continued . . .