Longman's definition of "corresponding" is
having similar qualities or a similar size, position etc to something else (SYN equivalent)
In other words, "correspondence" is a relationship between things that are similar, as in the example sentence from the Merriam-Webster definition:
"Robert" is a boy's name, and the corresponding name for a girl is "Roberta."
The sentence is about a direct relationship between two names which are the same type of thing. We use "respective" when there are several separate things and we don't want to list each relationship. For example,
Decisions about acceptable colors and markings for registration and Conformation competition are made by the national breed clubs for each of the respective breeds. (*)
The sentence above is about the relationship of each club with a different breed of dog, so we use "respective".
Here is another example where we would use "corresponding" and not "respective" because we are writing about the relationship between wires that have the same color:
One color at a time, twist the fixture wire to the corresponding color outlet wire and replace the wire nut. (*)
Here is another example where you would use "respective" and not "corresponding" because we are writing about a student and the dormitory they live in:
After the party, each student returned to their respective dormitory.
In your example, the power adapter and Ethernet cable are separate things that have different outlets, so "respective" is correct. If the cables and outlets were color coded, we could use "corresponding" because now they have something that is the same or similar - a color. For example,
Before you begin please connect your laptop's power adapter and an Ethernet cable to the outlet with the corresponding color.
