Why bother having different engine options on one aircraft?
When Lockheed built their L-1011 TriStar, they had one engine option, the Rolls-Royce RB211. RR did not deliver in time, and that has been attributed to the airplane's demise.

Although the TriStar's design schedule closely followed that of its competitor, Douglas beat Lockheed to market by a year due to delays in powerplant development.

So, having more than one engine option lowers the risk of a new airplane launch, and this has the benefit of allowing the airlines to pick their favorite.
Having one engine option on the other hand, like with the Boeing 777-300ER and the 777-200LR/F, allows cheaper certification, but it frustrates the customers:

Even if the 777X programme is successful, the sole-source deal has left some customers frustrated. Boeing argues that it chose the GE90 as the best engine for the 777X programme after its own rigorous competition. But as one airline executive says: &quot;Our response is we'd rather run our own competition.&quot;
He adds the suspicion that Boeing decided to take the benefits of the engine choice competition for itself rather than let it accrue to the customer. Also, by eliminating a choice of engines, the company saved the costs of certifying more than one of them, he points out.

Back to your points: Your final conclusion is right, the performance difference is not a factor. For the second point, airlines get maintenance deals, so they like to stick with the manufacturer they already have business with. Getting discounts is easier when there is competition.

Related and interesting: How much would it cost for a large airline to switch engine manufacturer for their planes?