I brand my pieces.
First, I sat down and came up with a logo. It is based on my initials, but looks more like a symbol. I then had a custom iron made. When applied to the bottom of a bowl the effect is very like the hallmarks seen on fine china or silver pieces. This ends up looking very personal in a way that my name in block letters would not.
The brand I made was about an inch in size and I usually place my mark centered on the bowl bottom.
My last project was a petite bowl with no non show surface. I realized the branding mark was going to be too distracting. I approached the branding iron company and got them to make me another as small as possible, about 3/8 inch.
Then as I continued the project I pondered that even though a discrete size, burning the mark was still going to be more distracting than I would like. I hit upon the idea of using the branding iron to emboss the wood surface.
After carefully positioning the branding iron head on the bowl I used a C-clamp to gently apply increasing pressure and ended up with a very nice and subtle embossed mark.
Pictures show my full brand on a larger bowl and the embossed mark on the little bowl.
I don't add any other info to the piece. I am thinking of numbering my pieces (ink), but for now I just have a catalogue of photos.
Here is a tip that works for me. Whether branding or embossing (and I think this advice applies to pyrography and maybe ink) I always apply my mark when I have finished shaping and sanding the bottom and before completeing the project.
My usual approach is to start turning the wood with the basin surface (top) mounted to the chuck and form the sides and bottom of the bowl. Then these surfaces get sanded. Now before flipping the mounting I apply the mark. In the case of hot branding I can use the lathe tool rest to help me align the hot iron (check the tool rest height before you heat the iron). Sometimes the brand gets poorly applied (even with practice), but since the bowl is still lathe mounted it's easy to sand or carve the mark off and get it right the second time. If the brand is good, lightly sand it to clean up the edges of the mark.
For embossing I have a similar sequence, but had to take the bowl out of the chuck to fit the C-clamp.