It appears that the built in browser doesn't allow you create actual polygons!
Did you see Rob’s response here? They are polygons in the database, even if they don’t show up that way in the editor. (Try looping a ‘Line’ back on itself like a snake eating its tail and you’ll see what I mean: it’s a different type of object, and you can’t give it area-type properties, such as ‘Allotments’. It will never be stored as a polygon, _vs_ an ‘Area’ which always will be.)
it creates a line with an internal buffer…
I’m afraid I don’t know what you mean here, but possibly this? Did you know that when in edit mode you can at least _display_ the object with ‘full fill’? While in edit mode, click the ‘spanner and gear’ icon on the menu on the right edge of your browser window and scroll to Fill Areas > Full Fill.
…such that the centre of an area doesn't register when clicked on.
As Rob says, clicking the very edge of a feature to select it is intended behaviour in the built-in editor, which is known as ‘iD’. I don’t have experience of the alternatives he suggests but they might be worth a look if clicking the centre of the polygon is indispensable behaviour for you.
Or might it be something that could be taught to proposed editors. Do you know of learnOSM.com and these tutorial videos? If you think a custom ‘for parishes’ video is helpful please let me know: I’d be happy to put my teacher/trainer hat on and collaborate.
the allotments in green dotted, not the allotment brown
‘The map’ is strictly the database: what you see at openstreetmap.org is just one of many ‘renderings’, this one known as ‘OSM Carto’. Green is what the OSM Carto developers have chosen for allotments, so that is how the ‘standard’ layer will always show them. On the other hand Andy Allen's ‘cycle map’ is a different rendering and that shows allotments as brown. There are umpteen other renderings – eg, here. (Where does the 'allotment brown' you refer to appear, btw?)
It’s well out of my experience, but I gather it’s not too difficult to customise your own rendering (=appearance) of the map (=the underlying data). If different colours is a big incentive for parishes to engage, it might be worth considering.
only the edges registering anyway
See above: the database knows this is a solid polygon, indicated by internal shading when you select 'Partial Fill' (which mere ‘lines’ don’t have).
seemingly unable to edit the green polygon that is there/unable to add a new polygon that isn't just a line buffer
If the issue is that you edit and then can’t see your changes on ‘the map’ (here I mean OSM Carto), this is likely because the image tiles take some time to update (and your browser cache can make it seem longer: it sometimes gives _you_ the version it saved days ago – but new visitors will see your edit).
Rest assured that once you press ‘save’ (the ‘upload’ icon top right in edit mode) and follow the workflow, your object is in the database – but you won’t see it until things update (see above).
(Other places on the web certainly give a swifter serotonin hit, and there’s a project to ‘gamify’ OSM to resolve that issue…)
Hope that helps
Feel free to talk here or on the forum (more heads more help) if you need further guidance - and again, fair winds and following seas for your effort to put OSM to work in such a useful way.