Can you provide links to the MW definitions and/or some examples of the contexts where you'd like to use these words?
They can have similar meanings, but I don't use them interchangeably. A commitment typically means that you have (agreed to) an obligation.
Colleague A: Can you come to a meeting at 2 p.m. today?
Colleague B: I'm sorry, I already have a commitment for that time. I have to meet with a client.
An engagement can also be an obligation, but in AE, it is most commonly used to refer to:
(a) the period of betrothal prior to marriage, or
(b) the act of having been hired for a specific project.
Colleague A: Has your new client signed an engagement letter yet?
Colleague B: No, we're still negotiating some of the terms.
Elisabetta