So, present continuous sounds more natural in this case, right? Because we are speaking about an action that is happening during this period of time. Right?

The difference comes from the verb phrases used in the sentence. -In "I'm from New York. I'm currently living in Chicago", "I'm" is actually the statement "I am". In this case, we are creating a present progressive statement. The equivalent of saying "is + verb+ ing". An easier way to understand it would be if the statement said "I currently am living in Chicago." The second sentence "I currently live in Chicago" is a simple present tense verb and thus does not require the "-ing" In it's basic form, the first sentence is more correct. If you wanted to make the second one sound natural, there would be a "but" between the sentences. "I'm from New York, but I live in Chicago right now". #1 is more natural sounding however.

I've heard these two variations even in teachers' videos on this site. 50/50