By all accounts, Bobo gave Maxim a whupping in their first fight, and knocked him down (more than once, I believe). He won impressively enough that it convinced many people that he could take on Archie Moore - but that proved to be disastrous. From what I've read of the Jones fight, Bobo was giving him a hard time through about five rounds (perhaps even beating him?), and then Jones found him with some good punches and KO'd him.
This seems to be about right for the Maxim fight, off what I have read. I saw the fight with Jones on TV. I have not seen a film since. You described it about how I remember it. Olson held his own for five rounds. Jones caught him with a right in the 6th and Olson stumbled back after being hit, fell against the ropes, and tried to pull himself up by grabbing the rope, but couldn't beat the count.
Here is a write-up on Olson-Jones from Boxrec. Scartissue 1960-08-31 : Doug Jones 176lbs beat Bobo Olson 179lbs by KO in round 6 of 10 Location: Chicago Stadium, Chicago, Illinois, USA Referee: Bernie Weismann 23-25 Judge: Jim McManus 23-24 "Doug Jones, 176, scored his 16th triumph without defeat when he kayoed Bobo Olson in 1:30 of the 6th round. Jones tossed a short left and a hard right uppercut, ending the scheduled 10 round Chicago Stadium light heavyweight feature. Olson, 179, had piled up a slight margin with an effective left hook which Jones didn't seem able to solve. There were no other knockdowns and neither fighter was hurt when the end came." -Associated Press Post fight comments "We'll just have to lick our wounds. I don't know what wounds we'll lick though, because Bobo wasn't even scratched." -Sid Flaherty, Olson's manager "I walked into one lucky punch. It was a right, the only one he threw all night." -Bobo Olson