Look for the best behavior practices for "write forward" and "close matches"

I'm working on a user directory widget for a B2B application. The search function is the most important part of the widget. We are currently showing the results of the advanced writing after 2 characters (therefore, the user types "bo", the results shown can be Bob Gundersonn and Sarah Flamboe). This is fine, but we are having some disagreements about how to handle close matches or fuzzy search.

Allowing close matches would help if you wrote Steven but you refer to Stephen. But it could be confusing if they write to Jennifer Barker, who has not been added to the system and needs to be created, and see results like Jian Jan and Jeremy Simon. (which we are currently receiving).

If there was not an advance writing functionality, you could wait for the user to enter your formal search and say: "There were no coincidences, you meant …". But it is a little more complicated with the type that is ahead, because the possible results that are being generated are being written, and that list changes dynamically and quickly.

What do you think is the best solution for this behavior?