I own all my cylinders. From actual welding cylinders to dive (O2 welding, O2 aviator, Acetylene, CO2, Mig mix, He). Over the years I have done business with different gas suppliers. Never once was I asked for any paperwork. There was one nice CO2 bottle I used off-roading that had a custom paint job I kept and that was the only one I ever paid for a hydro on. Bit of a pain as I would have to drop it off, come back later after they pumped CO2. They special marked it as customer owned. All the others were exchanged in or out of hydro without question. Just drop off at the gate outside, they take it in. The bigger place will radio in what I dropped off, the smaller places will set it aside and walk me in and do the paperwork. Pay the piper, and they will walk me out (often to pick the cylinder I want) or the man with the radio will have it at the fence waiting for me. I bought cylinders from friends, ads, whatever. It is a core that I turn in for a different cylinder filled with what I want.
I have paid upgrade charges. My little acetylene bottle could not keep up with the demands of the rosebud tip. Took the bottle in, they credited me the cost of the little bottle and charged me the price of the bigger bottle plus the fill. Did the same when I dropped off a little O2 welding bottle and exchanged it for a bigger aviator bottle. Other than the upfront cost I don't see value in rent/lease a bottle over owning one. The acetylene bottle I traded in had sat in the garage for a decade before going in. At $5 a month I would have paid for the bottle twice in those years. There were no added long term costs with owning either. If I were to rent something it would be short term. I could see traveling someplace and renting an O2 cylinder for a month and turning it in when done. I know job sites that rent because they are short term use. Or budget does not exist for capitol purchases but does exist for consumables. I do see reasons to rent. But I see the long term benefits of owning are staying in my future.