I've dealt with this issue many times over my career, so let me try to un-muddy the waters...
My reference is always some version of the Chart linked below, and I've used 'ohms/1,000' for any given wire size. It's easy to get your head around that length and ohmage for mental math.
http://www.powerstream.com/Wire_Size.htm
Yes, 1,000' of #18 is >6 times the resistance of #10. While that's true of even short wire runs, it takes 1,000' of wire for this difference to be even a few ohms total. The short run of #18 wire likely will be inconsequential in the real world. Yes, a short bit of #18 will act as a 'fuse'
relative to the #10, but only during long periods at maximum amperage [current] that #10 wire can carry, especially if the #18 run cannot air cool itself.
I used to advise folks making long, unwieldy wire/cable runs to run wires in parallel for any given size. You could do the same, and parallel another #18 or larger wire to the existing #18 wire. This trick would make the effective resistance of this short run ~half of what it was, or ~3 ohms [if it were #15 wire 1,000' long].
It's important to realize that resistance is evenly distributed along the length of any wire. That resistance difference is not all 'crunched' disproportionately into the short run you show in your very helpful pix. 'Run the numbers' to see #10 vs. #18 wire resistance in the relative lengths you're using.
The short length of wire on the connector ground side was because the expectation is that is goes to vehicle ground closely adjacent, or it grounds through mounting screws. This is not the case in a wood frame TD. It is assumed that the positive lead, +12 VDC run will be much longer, and more prone to resistance-related voltage drop, than a short ground connection made adjacent.
What you've got are called 'spade lugs' in the world of us 'sparkies'. You could buy a couple more spade lugs, and crimp on a larger size wire. It would then slip on the 'male' connector 'paddle'. Spade lugs come in various sizes, so take in what you have to match it in size [width].
I crimp and solder everything and put heat shrink tubing around it. Alternatively, connections should always be mechanically sound [crimped or twisted] before you solder. Wires should be firmly twisted together well before a wirenut is put on to withstand road vibration. Old timers put electrical tape around secured wirenuts as well.
Last, I've had Cig Lighter plugs themselves overheat and melt. Resistance is proportional to contact surface area. The rounded end of the Cig Lighter tip is WAY too small to carry the amps req'd in many applications. In what is called 'the thermal runaway effect', the tip of the connector corrodes or degrades as it heats. The resistance of it then increases. More voltage drop thus occurs across it, generating more heat, which corrodes it more. And so on... This is why Honda and other Generators use 'blade' plugs at unique angles to carry even 8 amps @ +12 VDC. Blade plugs distribute current over more surface area, overcoming resistance and thermal runaway inherent to small contact area. My point: there's other safety 'weak links' in about any Cig Lighter connection used to carry 'high' amperage vs. worry over a lil run of #18 wire.
Plug a 'high' amperage draw Accessory into your completed plug and hold the #18 wire in your hand to see if it heats up 'excessively' to you. That real world test will trump all theory. I think the same way, so I appreciate your viewpoint. There
is a hypen in A-R! ;-0