No. Both workers have created a sprocket, which exists as a direct result of their actions
I get the sense I've touched a nerve here. Though I'm not sure where.
I would like to mention I never suggested the frustration comes from workers not being able to own anything at all. Though I still believe what you said about workers not being entitled to the direct consequences of their labor is flawed. Similar to someone saying "I spend all day pulling oil out of the ground, but I'm not allowed to own any of it". That's a fair complaint, and I'd suggest any oil rig worker who wants to get paid in oil should absolutely negotiate for that with their employer. What I was trying to explain is the "direct consequences of their labor" is the compensation they're paid for providing said labor. You, as a worker, sell your labor for a price, same as any other transaction. If you will, your "sprocket" in this situation is the labor you provide.
Oh, I agree there's plenty of pros and cons to talk about regarding a capitalist economy. There's also plenty to talk about regarding the form capitalism has taken in our global and local economies. However, that's for the other discussion OP mentioned. This one is only about defining what capitalism is.
To that end, and in regards to your critique about parasitic owners living off the labor of workers, I'd like to pose another question: Does a capitalist system require workers and employers?
I would say no. It's entirely feasible for everyone in a capitalist society to produce and trade their own sprockets. It would be up to each individual to either inherit a given trade from someone who's retiring, or else find something society needs that they can fulfil. Our current use of capitalism may largely contain organized businesses built on employer/worker relationships, but that's not a defining trait of the system.
The cost and price of sprockets is also not a defining trait of capitalism. You're free to set your own price on any sprocket you own. You could ask for a handful of dirt as payment, or you could ask for the literal moon. Is the moon a fair price for your sprocket? That's debatable, and largely decided by your consumers
Our cans of sweet potatoes have an alternate label on the back that says yams, so apparently not much. (This is in the USA)
I'll toss my two cents in
What is capitalism?
Capitalism is a form of economy where individuals are allowed to obtain, keep, and distribute capital as they see fit, so long as they have the means to do so.
What is capital
Capital is a combination of property and money. Property being the things you own, with money being a measure of potential property you don't yet own.
How does capitalism empower people to own what they produce?
It's at the core of the concept. Individuals are allowed to obtain, keep, and distribute capital as they see fit. You and you alone own your capital.
I see a lot of comments saying workers are not allowed to own what they produce. That their employer takes it from them. I feel this is flawed and possibly comes from a place of frustration. So let me ask this: What does an employed worker produce?
If that worker is a self employed craftsman making sprockets, the answer's clear. They produce sprockets. They can then go out and sell those sprockets for goods, services, or money as they see fit.
If that worker is employed by a sprocket making company; they still make sprockets, but that's not what they produce. They produce labor. Which they've chosen to sell to the sprocket company for money and/or other benefits. They may not care about sprockets themselves, don't go to sprocket conventions, and certainly don't want to deal with figuring out how to sell all the sprockets they're making. It's a better deal to sell your labor and use the profit you make off your employer to do the things that actually interest you. It's your employer's job to handle everything else.
Note that selling your labor is no different than selling capital. You set the price you're willing to accept, and your employers (who are your consumers) can accept that price or not. That doesn't mean you can set your price at any number and expect it to stick. Just like how the price of sprockets is dependent on consumer need and competitive prices, so is the price of your labor.
Bonus Questions!
Say you are an individual who sells something you create. Are you a capitalist?
If you also exist in a capitalist economy, then yes.
If you are the above person, can you exist in both capitalist society and one in which private property has been abolished?
No. Capitalism cannot exist if you don't have control over your own property.
Say you create and sell some product regularly (as above), but have more orders than you can fulfill alone. Is there any way to expand your operation and meet demand without using capitalist methods (such as hiring wage workers or selling your recipes / process to local franchisees for a cut of their proceeds, etc)?
Yes. Ideally you raise your prices so fewer people buy your product while you still make the same profit as if you were filling those extra orders. Alternatively, you can work to optimize your production methods to create more product in the same amount of time. Be it finding more efficient methods you can practice to make each product, or creating or purchasing equipment which can make each product faster. The balance between price and optimization is up to you.
Is the distinction between a worker cooperative and a more traditional business important? Why is the distinction important?
It's a term to describe a type of business. No different than "corporation" or "partnership" or "nonprofit", among others. The distinction's important in that there's value in being able to describe different types of businesses.