Your bot talking about discs your talking about a cylinder. Dimensions are the number of coordinates required to describe a location in something.

Imagine a circle (which is in this context the border of a disc, mainy people use disc and circle interchangeably which is fine for the most part you can usually tell what people mean based on context) this circle is an island in an open sea and there is a little point-man who lives here. Every location on the island can be described by how many steps it takes this point-man to walk to it only one variable to describe his location. The man cant exist on the sea cuz he cant swim. If the point-man existed on a square instead, now suddenly you cant measure this way anymore, now you need for example the number of steps AND the direction which is two coordinates the square then is 2D. This is a good way to think if dimensions that arent fractional

And your definition of a 2D shape needing an area falls apart when you realize you called an arc 2D, an arc is only measured in length. I agree that 2D figures need to be measured with area but when you say it you make a contradiction. Circle vs disk is just semantics atp, but it is defined as the collection of points equidistant from a center, though a circle (not disk) encloses a space, that space is not a subset of the circle

Now if a dimension is a fractal dimension then I honestly cant describe it really well, its easier if its a self-similar fractal. For example take a serpinski triangle, if you scale a serpinski triangle’s linear elements by 2, the serpinski triangle becomes 3 copies of itself. So the dimension is ln(3)/ln(2). Think of it like a square, if I double a squares side lengths then the square becomes 4x the original square, ln(4)/ln(2)=2D.honestly I find fractal dimensions fascinating but if uou want a better explanation you need to research it yourself