How does 受 work in this context (on the first article)?
The main parts of the sentence is 三文鱼(subject) 是(verb) 寿司种类(object).很受中国人欢迎的 and 中国很受欢迎的 are the modifier of 寿司种类. 

To me, the first sentence is better, because in the second sentence it repeats 中国 twice. 在中国，三文鱼是中国很受欢迎的寿司种类. It could be better to remove the second 中国. 在中国，三文鱼是很受欢迎的寿司种类. 

受 is short for 受到， meaning 'to get or getting'. 

The phrase 很受中国人欢迎的 means 'getting warmly welcomed by Chinese people'. 

As a whole, 三文鱼寿司是很受中国人欢迎的寿司种类 can be translated to Salmon sushi is the type which is warmly beloved/welcomed by Chinese people. 

The subject for 受 is 寿司种类. In fact, it's a passive voice here. That 寿司种类
is 受到中国人欢迎. This is the same as "寿司种类被中国人欢迎". Somehow, 受到 here is the nice way to express “被” I think.