To answer the first part of the question, let me quote the first sentence of the Cone description:
A cone extends in a direction you choose from its point of origin.
That is, the cone is defined by the direction you shot it in, which is the axis of the cone. As such, it makes the most sense to refer to the other properties of the cone in terms of the axis.
For the second part, I have two points. First, I would agree you've found a mathematical contradiction if you assume the maximum length is along the axis. In that case, the cone defines an isosceles triangle where the height equals the length of the base. In such a case, the two equal sides would of course need to be longer than the triangle's height. This necessitates the cone ending in a flat plane, rather than being a surface of revolution. This contradicts the presumption that the height/axis measurement is the longest, and makes my response to the first section somewhat nonsensical. As you point out, it would be more consistent from this view to measure the sides as the maximum.
On the other hand, the purpose here is to decide whether or not a given target is inside the cone. You may not be using the optional grid rules from DMG 250. If not, then DMG 249 advises
go with your gut.
If you are, then all of your math is rounded to the nearest square. In this case, I'd definitely advocate ending the cone in a plane. The main reason is because it's quick an easy to do and thus keeps the game flowing.
You may also be using some kind of exact placement system, ala Warhammer, where you physically measure the distance between pieces. In that case, the DMG offers no advice and I'd simply ask your DM to make a call.