I love this stuff! I grew up seeing it called engine-turning and I wasn’t familiar with the term Guilloche before now. This work is another example of a Wonderful Thing. I loved those videos too, thanks!
The US Mint (and those of some other countries) used to use a machine called a Janvier reducing engine or lathe, which was used to engrave master dies (“hubs”) for coin making. I imagine that it was kind of like a Guilloché lathe plus a pantograph. Other reducing lathes were used for the same purpose before the introduction of the Janvier engine.