Answer to my question - enable replica tracking (data management tools) for the dataset and overwrite the service, if it was not done before publishing.
For those who watch this video FYI - it is best to enable sync (and enable ''create a version for each downloaded map'') and enable replica tracking before publishing.
Also another note for first timers as myself - make sure that the basemap you are using for your enterprise web map that you want to take offline comes from AGOL basemap service. Steps for this here Use an Esri basemap offline . Or in other words - as always read the documentation carefully! Even if you think that 5 min video says it all and very clearly (it is just not possible to tell everything to everyone).
Cheers and good luck!
P.S. I managed to achieve all I wanted yesterday:
- versions worked great both offline and on;
- offline map areas on my device downloaded successfully - sync worked great;
- could easily (after some struggle) create map areas both via Web map settings and Field maps;
- one problem very weirdly field maps app did not allow me to add a new feature anywhere else besides my current location (even if I disable the Feature Actions Collect Here option in the Field Maps web app)
and today - I still haven't fixed my yesterdays problem AND now I cannot create a Map area either by Field Maps web app, nor via Web map Settings page. No problems with the device and offline area creation+download. Nothing has changed overnight. Love this.