microG is an Open Source replacement trying its best to mimic genuine Google services, enabling a lot of Google-dependent Apps to run without the genuine Google services being on the phone.
Open GApps offers installable packages of the genuine Google Apps and services in different extents called e.g. pico/nano/micro/mini/etc. …
Can be used in connection with Google’s device registration for the use of Google Apps and services on non-certified Android OSes (to not endanger a Google account with violating TOS) … https://www.google.com/android/uncertified/.
LineageOS comes without both of them by default, the user may install whatever.
(microG and genuine underlying Google services can’t be installed alongside each other because microG itself fakes to be those same Google services to the system. Genuine Google Apps however can be installed and they may work with microG, or not.)
LineageOS for microG is a LineageOS fork with microG built-in for easier setup and use of it.
/e/ is a further de-googled LineageOS with microG built-in.
So, my try (to be corrected by more knowledgable people ) …
AOSP - Android Open Source Project, “Android” if you want, the base of all Android OSes out there, maintained by Google.
Fairphone OS - AOSP + hardware vendor contributions + Fairphone customisations + Google certification + preinstalled Google Apps and services (basically what almost every Android smartphone vendor does), maintained by Fairphone
Fairphone Open OS - AOSP + hardware vendor contributions + Fairphone customisations, maintained by Fairphone (for the Fairphone 2)
LineageOS - AOSP + hardware vendor contributions depending on the device + LineageOS community customisations and launcher, maintained by the LineageOS community
LineageOS for microG - LineageOS fork + microG built-in in a deep way LineageOS chose to not allow easily (hence the fork), maintained by the LineageOS community, LineageOS for microG just takes LineageOS and automatically deep-integrates microG into it
/e/ - LineageOS fork + extra steps to de-google it + /e/ customisations, launcher and Apps, maintained by the e foundation with an official partnership with Fairphone in place (for the Fairphone 3/3+)
GSIs - Generic System Images, experimental Android builds supposed to run on all devices with Project Treble support without the need for device-specific adaptations, but it’s complicated and results vary much, maintained by individual developers and enthusiasts.