Actually, re-reading your first post, you mention that you have the same issue in iTunes as well as on your phone and that's caused a "light bulb" to turn on above my head. (For those who take everything literally; please don't. It's a colloquialism)
Consider the information I've laid out below; is this the issue you are experiencing? If not, can you explain what is happening, because it makes no sense to me? More importantly, does my confusion over what I see and interpret make sense to you? At this current time, I wonder whether it's just me that seems confused by this.
Looking at iTunes; in my iTunes (Windows OS), I too find all sorts of unwanted episodes in my list. Check the two views in my iTunes Podcasts Library. I've ringed in orange, the areas you should focus on. The rest is part of a broader picture, to provide context to the area I'm focusing on:

On the left hand side is a view named Show: Unplayed/My Episodes, while the right-hand side is a view named Show: Feed. The question is; how does one interpret these views and what is the difference between them?
As far as I'm concerned, since there is no option to show "played and unplayed episodes that I have downloaded", even though it's a view I might want (because they're taking up space on my computer), I'm going to assume (rightly or wrongly) that Unplayed/My Episodes actually means episodes that I have downloaded but not yet played, while the right-hand side shows the Feed, which is a view of all episodes that are available, regardless of whether I have them on my computer. Using this definition, I have no view for played episodes that I have not yet deleted.
However, if this was the case, why does the left side show anything with a cloud icon? The cloud means that it is not downloaded. The blue dot means that it has not been played. So although the episode Tranquility Drones 01 mixed by Mike G (in the feed Ambient Music Guide Podcast) is unplayed, it is not downloaded and therefore not "my episode". Just below that episode is an episode titled crawling (from a different feed). It is downloaded, but it is also played. So although it can be considered as "my episode", it's been played so once again, why is it in this view?
Now look at the right-hand side: this is a view of the Feed, so it shows every episode available from the Ambient Music Guide Podcast. This is displaying episodes I have played (so no blue dot) and deleted (so there is a cloud icon). This Feed view makes more sense, but still doesn't explain the left hand side view.
Now consider another Podcast series. In the pictures below, I have scrolled down from the pictures above, so the two left-hand sides are part of the same Unplayed/My Episodes view, while the two right-hand sides are part of the Feed view:

Once again, the left hand side shows episodes with cloud icons, so not downloaded and therefore not "my episodes". In the (left-hand side) ringed area, episodes 1196, 1199, 1200 & 1204 are all "Unplayed/My Episodes", so should be the only episodes in this view. I would like episodes 1197 & 1198 shown as well, since they are "my episodes". The rest (including outside the orange area) should not be there.
Now look at the right-hand side view: This shows episodes 1201, 1202 & 1203 in the orange area. My question is "what is the difference between those three episodes and episodes 1194, 1195, 1205 & 1206?" All seven episodes have been played and deleted from my iTunes, so why are 1200-1203 shown only in the Feed view?
Is it me? Can I just not see what Apple is doing?
As for the Podcast app on the iPhone and iPod, that's even more of a mystery to me than the desktop iTunes.