I think some of the labels on their map are the names of roads/paths, not just sections

Hello.
Are you basing these edits on the map image on their website?
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Published using OSMCha: https://osmcha.org/changesets/144490496

Changesets and OSM in general have no way to upload and host images.
if you can post it on slack to explain what you are referring to.

If you use Android phone, there's app called Vespucci. It is a little bit clunky and awkward to use, but it does have "make square" functionality.
If you use iPhone, I think GoMap!! app can make things square as well.

By "everyone" perhaps you mean locals to the neighborhood, because if you were to say that to me I'd have no idea what you are talking about. Take a look at osm.wiki/Names#Local_names_(loc_name)

Another note, when mapping features that supposed to be rectangular, you can right-click on them and select "Make Square" (shortcut key Q) to make its corner have perfect right angles. That makes your mapping look neater and closer to the real world shape.
Similar for round features, you can draw them very roughly with just 4 nodes placed around the perimeter, right-click and select Make Circle.

Hello.
Please note that "crosses=Riverside Drive" is invalid tag. We do not tag what street a crosswalk crosses. It is inferred by the map itself by the fact that crosswalk intersects Riverside Drive streets.
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Published using OSMCha: https://osmcha.org/changesets/144459720

Please note that crosswalks and sidewalks do NOT get names of their streets. They are unnamed features.(except for very few exceptions)
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Published using OSMCha: https://osmcha.org/changesets/144459546

Hello.
What are you trying to map here?
Also please note that all these tags are invalid and must be corrected:
bicycle access=yes
car access=no
embankment=yes
foot access=yes
foot access=yes
Please see access=* for the correct use of access tags.
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Published using OSMCha: https://osmcha.org/changesets/144459502

Similar to the previous edit, that this is an incorrect way to tag parking and driving lanes. Please see osm.wiki/Street_parking for information on how to tag such features.
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Published using OSMCha: https://osmcha.org/changesets/144459240

Please note that this is incorrect way to tag parking and driving lanes. Please see osm.wiki/Street_parking for information on how to tag such features.
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Published using OSMCha: https://osmcha.org/changesets/144459225

Hello.
"North bus stop garden" sounds like a name you came up with. Please don't invent names for features. It's perfectly fine for features not to have names. Perhaps it is more suitable to add that information into description= tag.
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Published using OSMCha: https://osmcha.org/changesets/144459164

Hello.
Perhaps instead of park it's better to tag it as landuse=grass or landuse=flowerbed ?
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Published using OSMCha: https://osmcha.org/changesets/144459116

Hello.
why did you delete building=yes tag here?
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Published using OSMCha: https://osmcha.org/changesets/144439602

Hello.
Please split your edits grouped into smaller geographic regions, such as one neighborhood at a time. Or a city at a time at most. There's no need for this edit to stretch from NYC to Stony Point.
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Published using OSMCha: https://osmcha.org/changesets/144420215

If a biz has multiple locations, such a chain or a franchise, it's helpful to also add branch= tag so help distinguish such locations.
If you know the official website you can easily get biz hours from there and then use websites such as https://www.webmapping.cyou/WebToOSMOH/ where you can simply copy/paste hours from the official site and it "should" be able to convert it into proper OSM tags. If that doesn't work, you can use https://projets.pavie.info/yohours/ to visually mark biz hours and get valid tag as well.
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Published using OSMCha: https://osmcha.org/changesets/144416383

Please don't fully delete closed biz, but instead mark it as discused, or at the very least delete biz-specific tags but leave address node intact. That way you are preserving history of this object if there's ever a need to refer to it in the future and you allow this address to be searchable and routable to.
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Published using OSMCha: https://osmcha.org/changesets/144415787

mapped it for you in changeset/144453136
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Published using OSMCha: https://osmcha.org/changesets/144415373

Hello.
When adding phone numbers they should include country dialing code (in US it's +1).
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Published using OSMCha: https://osmcha.org/changesets/144414902

It seems like you damaged flower shop tags in this edit?
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Published using OSMCha: https://osmcha.org/changesets/144393928

Hello.
Please note that when a business occupies only a part of a building, such a restaurant on the ground floor, it must be mapped as a separate node, not as a whole building.
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Published using OSMCha: https://osmcha.org/changesets/144393792