Can there be motion without the flowing of time?
Yes, in that "the flow of time" is a figure of speech. It is an abstract thing. There is no actual thing called time that's literally flowing. Instead light moves, planets move, people move, hearts move, blood moves, lungs move, air moves, electrochemical signals move in nerves and brains, and so on. Everything moves, including pendulums and cogs in clocks, and oscillating quartz crystals and electrons in the circuitry within the watch upon your wrist.
Can we reject the notion of the passage of time, but still accept that motion is real?
Yes, because you can't see this passage of time, but you can see things moving. This is the empirical ontological science of it. When you open up a clock you don't see time flowing through it. A clock is not like your gas meter. There is no time flowing through it.
My counter argument is that the definition of motion involves the passage of time. From wikipedia: In physics, motion is a change in position of an object with respect to time.
There's nothing wrong with that definition. Provided you remember Einstein's operational definition, and what a clock really does. It's a device that features a movement, wherein some kind of regular cyclical "local" motion, such as that of a pendulum or a rocker or a piezo-electric crystal, this motion being accumulated and/or converted to present some kind of analogue or digital display that we call the time. So motion is a change in position of an object with respect to the change in position of another. Or if you prefer, its gradual change in position.
Therefore, if we want to reject the notion of the passage of time, we must also reject motion.
I disagree. Don't reject what you can see with your own eyes because you're rejecting something you were never able to see.
If time does not flow, then it is like the spatial dimensions, and what our brains interpret as motion and time flow are really just fixed coordinates in spacetime.
No they are not. Those fixed coordinates are an abstract thing, and so is the "block universe". Ours is a world of space and motion. The map is not the territory.
Furthermore, that you can't see something doesn't prove that it does not exist. We cannot see an inverse square law, but we can deduce it from observation of things we can see. It is an abstract concept, but we accept it as reality (or at least until an alternative theory replaces it, but the issue here isn't whether or not today's science is correct).
We accept it as reality because forces really do diminish such that when you double the distance, the force is reduced by a factor of four. We refer to this as the inverse square law as convenient shorthand. But we understand why it applies, and nobody has an issue with it as far as I know.
My issue with this is that it appears to involve circular reasoning: 1. Motion is a change in position with respect to time. 2. The passage of time is not real, because when we observe a clock we are really just observing its cogs. 3. The cogs are undergoing motion - go back to line 1.
Note that I'm not saying time isn't real at all. I think it's best to think of it as something like heat. Heat is real, heat will burn you. But it's an emergent property, not something fundamental. Think about the kinetic theory of gases, wherein "the temperature of an ideal monatomic gas is proportional to the average kinetic energy of its atoms". The faster those atoms move, the hotter the gas. The temperature of that gas is akin to an average measure of motion:
GNUFDL image by Greg L, see Wikipedia
Time on the other hand is more like a cumulative measure of motion. And it's just as real as heat, in that a hundred years will kill you just as surely as a hundred degrees C. It just isn't something that literally flows or passes, or through which you can move. You don't literally climb to a higher temperature and in similar vein you don't literally move forward through time, and nor does time literally flow or pass.
So, who is right? Can we have motion, without the passage of time? If so, what is a more accurate definition of motion, which does not contain a circular reference to motion itself?
When you measure the motion of that object, you use a clock. And that clock really does feature other things, moving. That's the science of it. That's the physics of it.
NB: You might be interested in reading this popscience article I wrote a couple of years back explaining why time travel is science fiction. You "travel" to the future by not moving at all whilst everything else does.