Since the board has an odd number of intersections, without ko and seki, black will play the last dame. If two systems would to give the same result, either black needs to give 1 more komi, or force both sides to play equal amount of moves. Without the adjustment, they naturally give different results. In a sense, black always has “sente” for the final move. Ko changes the sente/gote is the reason why when one side cannot win and run out of ko threats, naturally has to pass.
In general, this technique it’s not about black gote move, but winning the ko throught the use of gote/dame move as ko threats (in Chinese rule, most dame is 1 point gote yose). If it is a larger gote value yose, then the other ko value need to be considered. In your example, winning the ko is only worth 1 points in Chinese rule, but C1/D1 is 2 points gote yose. In that case, black’s option is better to win the ko using that as a threat. Even if there are only dame moves left, as long as there is an odd number of dame moves left, they can be used as threats (since the last one is worth one more, think about the whole dame exchange process as one giant 2 point gote yose in Chinese rule).
This is where this technique gets its name from “(at the end) connect the ko (by being able to) play (the last) gote dame (as ko threats, when you have other larger and more ko threats left)” and get one more point as a result (under Chinese rule). In your first variation, black has one more ko threat at E2, so the result will be in Japanese rule B+3 (I originally type B+2, it is a typo), and in Chinese rule B+5 (black’s best action in Chinese rule is not to pass and let white win the ko and play the last move, but to keep the ko, as long as there is at least one larger threat left)