Lets keep in mind that all sports can be done for recreation but it doesn't make them less of a sport when thrown into the competitive context.
The two items I see people getting hung up on are
1. The competitive aspect of pinball
Being competitive against others is integral to the definition of a sport. No one can argue that pinball has no competitive aspect to it. Bringing up recreation as if it diminishes the sporting aspect is ignoring the fact that you can do *any* sport for fun and relaxation, and the recreational aspects of any activity don't diminish the sporting aspect in a competitive context. That part of the discussion should be mute at this point.
2. The physical aspect of pinball.
Most definitions of a sport only imply some level of physical exertion. While you can't sit on a counch and watch pinball play itself, you can sit on a stool at a machine and use physical reflexes to play the ball, and nudge a 300lbs box just enough to keep the ball in play but not enough to tilt.
Lets go over a few definitions:
"an activity involving physical exertion and skill in which an individual or team competes against another or others for entertainment."
"an athletic activity requiring skill or physical prowess and often of a competitive nature, as racing, baseball, tennis, golf, bowling, wrestling, boxing, hunting, fishing, etc."
"a contest or game in which people do certain physical activities according to a specific set of rules and compete against each other"
"a game, competition, or similar activity, done for enjoyment or as a job, that takes physical effort and skill and is played or done by following particular rules"
A more interesting conversation is when we cross over into video games as a sport. Pinball can be virtualized so that you can sit at a couch and the need to nudge a 300lbs box turns into a press of a button. Lets start by saying not all video games are competitive, a lot are just guided tours. Good examples are multiplayer team shooters, where the software acts as a medium for person vs person combat. A board game (chess, checkers, risk, etc) may involve strategy and competition, but they don't require any reflexes, so are classified as games. Back to pinball... it requires top notch physical reflexes to be a top player it's tough to argue it is not a sport by any definition. In that one can probably argue that multiplayer team shooters are also sporting activities.
The lines are already starting to get blurred... in the past we were used to only seeing sports like baseball, basketball, football, hockey, soccer, boxing filling arenas and convention centers with competitive participants and fans, but now we are seeing competitive video gaming and pinball filling similar venues... and the continued development of AI and robotics are ushering in an even more grey area of competitive sporting events. Drone wars, robot wars, they are physical and they are competitive.