how to get podcasts to sync from ipod to itunes

A number of iTunes versions and iOS versions ago, podcast management seemed very simple. I was able to download podcasts on my computer, sync unplayed podcasts to my iPod touch, then next time I synced, it would remove the played episodes from the iPod but keep them on the computer. This is what I want to happen, but add the ability to download podcasts directly on my iPod, then have those episodes sync to my computer.

Now, everything is so confusing. There are two podcast tabs on my iPod: My Episodes and Feed. What is the difference? No idea. Feed seems to include undownloaded episodes, but sometimes so does My Episodes. If I download while in the Feed tab, sometimes it shows up in My Episodes, sometimes it doesn't. Something similar is in iTunes, but there are so many tabs and dropdown menus, I don't know what means what, and why very old episodes seem to disappear from my computer, why some are unplayed in the Feed but played in My Episodes or vise versa. I have the setting to keep played episodes selected, so they shouldn't be getting deleted just because I've played them already.

Could someone please help me? To put it all together, ALL I WANT when I sync is for unplayed episodes to be on my iPod, and played episodes to be removed from my iPod and stay on my computer. I want this regardless of whether the episode was downloaded on my computer or on my iPod. Is this even possible?