Yeah, so a country is generally considered to be conquered after it surrenders. Any other definition would be pointless. After all, you can say that Song China wasn't really conquered: the imperial government surrendered in 1276 before the last remnants were defeated in 1279. Or, for a more modern example, France in 1940 wasn't conquered by Germany either: the civilian government of Petain was the entity which sought the armistice. Same for Japan in 1945: it was a civilian, the Emperor, who overruled a military that wanted to continue fighting.
True, but all of these examples, except for the last one, involved both sides undergoing military conflict, and one side managed to gradually overrun and retain most of the conquered territory until they were decisively pushed out years or decades later. Both the Mongols and the Nazis continued to push into foreign territory and continued to occupy the vast majority of it until they were forced to retreat altogether in 1368 and 1944, respectively. On the other hand, although the Home Islands were never invaded by either the United States or the Soviet Union, the former continued to actively occupy major cities for seven years until a treaty was finally implemented.
In comparison, the Mongols withdrew their troops from Korea after each campaign, although some garrisons were retained in the north and on Jeju Island for decades. However, the capital and most of the peninsula were never occupied by the invaders after 1273, although Mongol diplomats continued to influence politics within Goryeo. The fact that both Yuan and Goryeo rulers took spouses from the other state for decades also suggests that the relationship between the two were set apart from other ones.
I really won't speak about Georgian history, but my point is that Koryo wasn't exceptional in retaining its existence despite surrendering. The Rus states were already splintered even before the Mongols came (I think there was a nominal unity under one of the rulers, but functionally they were separate), and even after the Mongols invaded, they still continued to exist.
See above regarding Georgia. The point about the Rus suggests that although each polity was relatively consolidated, and had close ties with each other, there was no single ruler who dominated the others. This essentially suggests that a relatively centralized state technically did not last both shortly before and after the invasions by the Mongols.
Also, even though Vietnam and Champa paid tribute to the Mongol Yuan Dynasty, they were essentially independent, unlike Koryo, whose rulers could be deposed at will by the Yuan Emperors. I don't even think the Mongols even reached Champa. Like Novgorod and some of the Rus states, it paid tribute as an insurance measure against war, not because it had to.
Yes, but the two situations are different, as Goryeo continued to resist until the last remnants were finally defeated in 1273, while both Vietnam and Champa decided to send tribute and nominally surrender even though the former had scored three victories. Goryeo didn't have a choice when it finally surrendered, while Vietnam and/or Champa could have theoretically continued to maintain an aggressive stance even after the third invasion, although they eventually chose peace for practical reasons.