MapBBCodeShare VS uMap

I like using http://share.mapbbcode.org for a similar purpose.

Umap?

If that's not feasible and what you're doing isn't secret then perhaps something like uMap might work? This allows you to add overlays for the things you want to highlight and has a few different styles of basemap that tone down some of the clutter of the default "default" style. It also lets you pull (small amounts) of data into it via Overpass e.g. pre-mapped fishing spots or toilets. This can then be shared as a link or embedded in a site.

Having an OSM basemap overlaid with points is easily possible with uMap, for example https://umap.openstreetmap.fr/

Have you used the OpenStreetMap ecosystem of ID/JSOM or maybe uMap (https://wiki.openstreetmap.org/wiki/UMap) if you need the data separated from OSM?

Not an App, but did you take a look at https://wiki.openstreetmap.org/wiki/UMap? With it you can make personal maps with OSM base layers.

Depends what you mean by private. For something shareable on the web you might want to try one of the uMap instances, not sure if you can keep them secret though. If you decide to use them I recommend you log into an OSM account first as otherwise it is very easy to lose the edit link.

For public display one of the uMap instances is probably easiest. It should allow you to import your previously exported points and also allows line and area attributes. The share link included embed code for a website. The instance linked by /u/beardy64 is probably the most popular, but there are several listed at my previous link. I would recommend you log in before creating the maps as without a login you have to carefully keep track of you edit URL's if you might ever want to update things.

- https://github.com/umap-project/umap/blob/master/docs/install.md

If someone is looking for an open-source webinterface for OpenStreetMap that offers a bit more eye candy, I recommend taking a look at Qwant Maps. Another useful OSM tool is uMap (source).