Thank you @screenager, your experience is extremely helpful to me.
Another test with microG is GmsCore 0.2.13.203915 on a third device shows a different result: the option ‘Exposure Notifications’ is displayed.
First I installed on a Xiaomi Redmi Note 4 “mido” this latest LOS-ROM: lineage-16.0-20201022-UNOFFICIAL-microG-mido including preinstalled microG GmsCore v0.2.12.203315 from the official LineageOS for microG website.
As Covid-19 app reference I use the German RKI Corona Warn App. It’s known for its proper functionality, its privacy friendliness (for this the Chaos Computer Club e. V. (CCC) has among others) and last but not least the German main developer of microG, Marvin Wißfeld, is interested in the app working with his microG system.
As a second test app I use UK NHS Covid-19 App to check the camera option “venue check-in” which does not work in all versions of microG GmsCore so far.
The installation of OS and Covid-19 apps was done as usual without any additional changes to the default settings.
Conclusions:
The option ‘Exposure Notifications’ is available in both microG versions: GmsCore v0.2.12.203315 + GmsCore 0.2.13.203915. Both Covid-19 apps work with both microG is GmsCore versions.
So I have to say that if you install the Covid-19 apps the same way on different /e/ OS and LOS, the results will be different.
On my two unofficial /e/ OS e-0.13-p [pʰeː]) + /e/ OS e-0.13-q [ kveː], the option ‘Exposure Notifications’ is not displayed and an activation was not possible with until now.
The current (16.11.2020) unofficial mido build from the official LineageOs-for-mircoG website works as described by other forum participants.
By the way, my question remains: Are there two different versions of the current microG GmsCore v0.2.13.203915 or how should I explain all this?