I've created this example search field, with toggles on each end (the dots/ellipses) allowing you to toggle between 'starts with' and 'ends with', because I was hoping to stay away from the ideas in this UX.SE question and its answers, for the reasons mentioned in the question, as well as the comments on the answer that suggests auto-complete functionality.
However, besides my example not really being finished yet, I'm far from convinced this is clear/intuitive enough. Surely though, there must be a concise way to make something similar like this work. Do you have any ideas how to improve this search field, to allow for easily toggling between 'starts with', 'ends with', 'contains' and 'equals', without cluttering up the real-estate and without the need for users to use input characters (regular expression-like and/or SQL-like syntax, etc.)?
Here is a very rudimentary mock-up of the envisioned UI: As you can see it's a very basic UI, much like any other data table/database-style application. The 'search:' row is where the intended fields are supposed to come. There's probably going to be one or two additional columns ('created' and 'modified') as well, which will have date-range search fields.
Here's a second iteration, based on @AndrewMartin's astute remark that there appeared to be a rendering bug, because of the missing side-borders. I have to say, I have the feeling it's slowly going somewhere now, but I'd still much appreciate your feedback/additional input/alternative ideas.
I've created a different concept with a cogwheel that shows the options on hover (which looks good in Firefox, but terrible in other browsers, I just realized), inspired by @Alvaro's suggestions. I'm still curious to hear your feedback and/or alternative ideas.