Well, I develop commercial application which exports selected rectangle of map as image. On frontend I use Leaflet (with esri-leaflet), and on backend I do simple C++ get requests to REST API export services. I mostly use arcgisonline public services like
Why do you find the answer right after you ask a question? Oh, because Search is awful but Related can save the day, huh.
https://enterprise.arcgis.com/en/server/latest/administer/windows/about-arcgis-tokens.htm
Anyway, still waiting for a proper answer.
Turns out, to export images you should use (for Export) variants of base maps. Like this:
Did they mention it somewhere? Nope. How did I found? A wild @esri appeared and replied on random comment.