Weirdly enough I just had this done in my place. Word of advice, replace as much of the copper as you can in that area. If you have 1 leak now theres a good chance another one along that line will pop up not long after. I should know since I had about 5 leaks in 10 years. We finally ripped out all the walls so we could do the entire run to prevent any new ones again. The link you sent was exactly what we used to link the existing pipe where we cut to the new PVC pipe put in its place. Just make sure both pieces fit snug together through the connector before tightening and thats it. To cut the pipe out we just used a sawzall with proper blade to cut through it.

I really didnt know much about any of this so most of this was from the help of my father who pretty much does everything lol. So I hope my advice is correct as he is not a plumber but just someone who does everything. With that said he has done 2 full remodels of houses basically himself and passed inspections on them without issues so I do trust his advice.