Do your neighbours not have any buttercups at all or do you not see the flowers? Just wondering if they need longer than a week to open new flowers if others are mown off.
Have any of the lawns had week killer applied over the years? At a guess I'd have thought clover may have been killed off if previous owners had used weed killer.
Do neighbours water their lawns?
Typically a close mowing favors grasses as they tolerate it better (you are still leaving photosynthetic leaves behind, and new leaves have already started to emerge before you mowed). Some broadleaved plants have also adapted to close haircuts, like buttercups (or dandelions, ground ivy, etc) White clover tolerates close cutting better than any other clovers because of its growth habit, but you have to have a white clover that grows shorter than the lawn gets mowed, otherwise you end up with stems that have no leaves, which is clearly a competitive disadvantage for the clover.
So, yes, mowing height can definitely have an influence on the species present. But you'll only see species emerge that are already there to do so. As said above, you may need to buy some clover seed. If you like to mow low, make sure you get a dutch white clover variety (not actually necessarily seed from the netherlands, that's just the descriptor of shorter varieties. Someone like Rob R might buy a ladino type clover variety for their pasture because they are taller and yield more biomass).
You also might need to pull established buttercups if you want to lessen their presence...
Another thing to check on is the pH of your lawn soil. legumes won't tolerate much acidity at all, and cannot chemically fix nitrogen if they aren't in the correct pH.
AFAIK none of the lawns has had any attention other than mowing since I moved here eight years ago. I think it's unlikely that any of then has ever had more TLC than mowing and occasional raking.
My lawn also harbours many other plants and the same is probably true of the neighbour's patches, but it's the daisies, buttercups and clover than are most noticeable.
I'll take note next time my lawn is cut how long it takes for the buttercups to flower - thanks for that idea TD.
My older neighbours sometimes leave it longer between cuts when the clover is most prolific because the bees love it. I'll try to remember to look out for buttercups when they do that.
None of us water the lawns.
Slim, thanks for your suggestions, but I'm just observing what's there and wondering about the differences between the three adjacent gardens. I don't have any choice about how my lawn is cut. My landlord does that task and I entertain myself from time to time by taking note of what is growing there. A few years ago I audited both my lawns and found the following plants:
Buttercups (back garden only)
Lesser yellow trefoil (lots of this)
English Plantain (many more in front garden than back)
Geum (back garden only)
Thistle (back garden only)
Field chickweed (back garden only)
Liverwort (back garden only)
Forget-me-not (self-seeded from garden plants, back garden only)
Viola (self-seeded from garden plants, back garden only)
White clover (front garden only)
Self heal (front garden only)
Bittercress (front garden only)
Mind your own business (front garden only)
Common mallow (front garden only)
The back lawns occupy a very small area in total - two gardens are about 20' wide x 30' long and the third is about 30' sq separated by 5 or 6' wooden fences.