I currently utilize the standard Umbraco Examine Search, which provides exact matches for up to 3 characters and is case-sensitive. From 4 characters onwards, it also returns results when part of the search query appears within a word and becomes case-insensitive. However, in our scenario, it's imperative to modify this logic to accommodate searches from 3 characters. How can this adjustment be made?

you are able to override the examine indexing options.. maybe you can create your own

CustomFullTextType

based on the one you've noted above and override on the fields you require? We've used this approach to override the analyzer and protected content inclusion.