How to respond to fuzzy requests such as "work more independently"?

I have a part time job where I work directly with the owner. He said he wants/would give me full time hours. Today he said that for this to happen he wants me to work more independently. I told him I found that a bit surprising, and listed 3 recent tasks I completed independently. He replied he helped with one. I replied he wanted to but I would've been happy to do it all by myself. What's a constructive way to interpret rather nebulous feedback like work more independently? Should I follow up with an email to ask if he could be more specific in what he wants or give examples of where he wished I hadn't asked him something? He said he wants me to be able to take more of the work load off of him, and by that I think he means I need to be on boarded into more systems and receive training for them. This isn't something I can do by myself. Should I send an email suggesting we create some sort of plan for on boarding me into the needed systems?

I think the most important thing to consider is precisely what you want out of this situation, and you haven't communicated that with us. You seem more focused on the communication of the owner, but I'm going to switch gears and assume that your concern is more about getting switched to full time.

Forget all this stuff he's saying about "work more independently" and then how he wants to argue (yes) with you when you point out how you've been doing it already. That's unproductive. I would be annoyed too, but again, it's time to shift gears.

Figure out what this guy's big pain points are in operating the business, specifically which ones are the most time consuming. From those, decide which you can handle without having to stretch yourself too far. You need to be able to "hit the ground running" on whichever of those things you come up with. Then, ask directly if you can take care of those tasks. The time consideration involved might be obvious, or you might have to mention that you're going to exceed whatever part-time limit you currently have now.

If you are approved to do the work past your part-time limit, I strongly suggest that you document whatever was stated in the conversation by any means, and be sure that your documentation is available away from the guy's business offices (it could be a personal journal, sending an email to yourself in a personal account, Google Docs / Office 365 online, or whatever). The reason for this is that the guy sounds kind of flaky, and you don't need any of that when it's time to get paid. Keep good records until things smooth out -- a document may help to resolve disputes later.

If you are a 1099 contractor (meaning you haven't filled out a W-4 form, I-9, and the guy doesn't deduct state/fed taxes from your pay) do not increase your hours without having a written contract in place for your services, that at a minimum states your hourly rate. (You've been warned!) Deal with the increase in hours after that.

Good luck.