Just my opinion here:
I once had a branding iron, and like others, found it to be finicky to use properly. If it wasn't used on a flat surface, it's a no-go. If it's not held perfectly perpendicular to that flat surface, it looks sloppy.
Yes, you can use a Sharpie, and many turners do. It's very easy to use, very distinctly black, and the lettering looks cleanly made.......but, IMHO, it looks very unprofessional.
Also, many artists and craftsmen are putting a multitude of information on their creations.....like name, date, production #, species, made in xxxxx, etc. To me, anything other than a signature, or personal logo, looks "touristy" and catering to a specific mentality level of a potential buyer who is influenced differently than the true connoisseurs of art.
For awhile, I attempted to use a woodburning tool to do a complete signature, and add things like species, date, etc.......but, I found that it was just too difficult to get clean looking writing on different kinds of woods. There is no second try, if you screw it up!
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What I've found works best for me, is this CSUSA "cub" wood burner. (They look a lot different now, but the internals haven't changed. This one is 30+ years old, and still works perfectly.) A logo is fine, and that's what I choose to use. Notice that my logo is designed with all straight lines.....That makes it very easy to get nice cleanly done results. Curved lines are difficult to burn cleanly. The real trick here, is to get the heat setting right, and go slow and deliberately. The heat setting will be hotter with more hard/dense woods, and less hot with softer woods.
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