The explanation is correct, I agree with the interpretation. Note the difference in the case of 'Reise' of Akkusativ vs. Dativ - that makes the whole difference:
Jemanden auf eine Reise mitnehmen
--> to take someone on a trip / to travel together with sb.
This usually means that the whole trip is done together. 'Reise' is in Akkusativ.
Jemanden oder etwas auf / während einer Reise mitnehmen
--> to pickup sb. or sth. during a trip
'Reise' is in Dativ case. This refers to something picked up during the trip (and not necessarily the rest of it either). It can mean you just picked up that person one day during travel and parted ways later.
As to a good reference explaining this difference... I sadly don't have any either. Note though, that the meaning of 'mitnehmen' is the same (approx. 'take along'), the main meaning (Hauptform) as noted in dwds. The difference is mostly in 'auf die Reise' (Akkusativ: for the whole trip) vs. 'auf der Reise' (Dativ: during the trip).