Simple answer: mainly the former. I use language that differentiates the two during training, because I want clients to start thinking through their canine companion’s perspective. It’s not about whether they are giving something to their dog for behavior; it’s whether the dog desires it enough to motivate future behavior. By using language that distinguishes the two, I’m influencing the way they process information during training, which affects their meaning-making systems. People do what is meaningful, and they avoid what is meaningless. Utilizing reinforces becomes meaningful, and giving rewards becomes (potentially) meaningless (and at the very least less meaningful). When it comes to the dynamics of meaning, change the language and you change the game.
Complex answer: sort of both. They are two different functional descriptions and I prefer to keep them separate to emphasize the contextual dynamics. If I give a dog a food treat as a “thank you” for behavior, then it is a reward. It doesn’t have to increase future behavior. If I am able to use it to increase future behavior, then it is an identifiable reinforcer. It only remains a reinforcer as long as it remains desirable. If I keep feeding a dog food because it has been a proven reinforcer, but the dog becomes full and doesn’t want any more, then getting the dog to take the tidbit when s/he doesn’t want it, then the reward can actually become aversive (which moves into punishment, which decreases future behavior). So, the reward that was an identifiable reinforcer can become a punisher if used incorrectly.