Why do you need a thicker wire when wire length increases?
The wire gauge you need is a function of several things. 


The acceptable voltage drop or power loss (that appears to be the
only thing considered in the website you linked). The voltage drop (and power loss) is proportional to wire length and inversely
proportional to the cross-sectional area of the wire- in other words
inversely proportional to the square of the wire diameter (assuming constant current).
The acceptable temperature rise. This is a function of the number
of current-carrying wires bundled together, the environment (maximum ambient temperature
and air pressure or altitude, for example), the insulation type, the
wire type (some types of wire are plated to withstand higher
temperatures than bare copper without corroding).
Regulatory requirements and other considerations- for example, the
wire may be rated for 200°C insulation, but you might not want the
wire to run that hot.
Fusing- the fuse or circuit breaker should protect the wire in the
case of faults such as overload or short circuit.


Very short lengths of wire can depend on heat sinking through the ends (indeed, in a vacuum, that may be the main heat loss mechanism), but usually that's not taken into account. 

Normally you'd run through a checklist such as the above to make sure ALL the requirements are satisfied simultaneously, so you might find that using PTFE insulated wire allows you to use AWG 18 wire, but because of the voltage drop limitation you'll have to use AWG 12 wire.