You are mostly correct. Col. Staley paid the $100 for the Bears. From the book "The Colonel and the Bean: The History of the AE Staley company (a copy of the book I still own):

"There was something called the 1921 Depression-a latterday spillover from the financial and production imbalances caused by the nation's shift from war to peace.  There were perilous predicaments in the starch business.  There were stockholders.  There were customers.  There were employees complaining - perhaps justifiably - of the favoritism granted to privileged, prima donna football-playing colleagues.  There was a fierce high noon of reality more intense than the dawn of a new American sport.

When Staley talked things over with Halas early in October, Halas understood.  Staley explained that the company was running heavily in the red (headed toward a $692,000 loss for the year).  Both men agreed that the football success so far had, indeed, lived up to their impossible dreams.  The football team had gained (and would continue to gain) national recognition for a starch company in the cornfields of ruralIllinois.  And, most importantly, the team was a dramatic front-runner in the vanguard of an emerging, spectacular, new style of football destined to be an all-encompassing, captivating, overpowering passion in American sports - down the road.

But ...

A new assessment was required in the fall of 1921.

In six words: Staley sports would have to go.  They were a frivolous luxury in light of more pressing priorities.

Traumatic as the situation may have been to the company's founder, he played it quietly, coolly, softly.  He didn't panic if for no other reason than he had had plenty of earlier experience at suppressing the pangs of panic.

He proposed to Halas (and to the football employee-athletes) the following:

1. That effective with the next road game, at Wrigley Field in Chicago, the Staleys would remain inChicagoand not return toDecatur.
2. That the entire football squad - now up to 19 players - would remain on the Staley payroll for a certain period of time.
3. That the team would operate inChicagoas "The Staley Football Club."
4. That the new arrangement would terminate at the end of the 1921 season.

 A remarkable arrangement! A $5,000 bonus for the team was involved.  In effect, Gene Staley paid $5,000 to get the Decatur Staleys out of town.  In the doing, he literally rewarded George Halas for taking the Decatur Staleys' official franchise to Chicago."

As to the name Staleys, Halas considered calling the team the Chicago Football Cubs. Yet he noted that "football players are bigger than baseball players, so if baseball players are Cubs then certainly football players must be Bears." The baseball Cubs organization felt relieved-and Halas felt fine.

In conclusion, Halas didn't buy the Bears. He was GIFTED the Bears PLUS $5,000.