Question: I think it originates from MMORPGs, particularly World of Warcraft. I think it refers to more than one thing. Did it really originate from MMORPGs? Does it mean one thing exclusively, or can it mean something else ina different context?

What does the term DPS mean?

Answer: 
Yes it is an appropriate use of the term. It is more often used to refer specific damage output per second in the literal sense. But classes that are known for specifically dealing damage can be referred to as DPS. It can be used similarly to how a class would be known as a healer or tank class. 

Most classes in most games deal damage and would have a DPS but as you used the term, most games have classes that are specifically used to deal significantly more damage than others. Those can be referred to as DPS classes in many circles.
"Damage Per Second". DPS refers to heroes whose main purpose is dealing damage. This (generally) means offense characters. Alternatively, it is the ability to put a lot of damage on a single target, allowing for quick picks. 

(Example) In Overwatch, a pharah can dish out insane DPS due to splash damage, no fall off on her rockets and her ability to focus on targets other team members can't. Meanwhile, a hero like Mei does significantly less DPS.