Do you see how much easier to build the second design will be compared to the first design. They will both do the same job which is to hold a shaft while it is turning. The most simple base I can think of is a piece of 1/4 inch x 1 1/2 inch aluminum bar stock about 2 inches long. Drill one or two holes into the bar stock to attach it to the lathe bed and have another hole to insert a 1/4 inch rod or ground dowel pin. The rod will connect the base to the round piece. The round piece will have to have a hole drilled into it to accept the rod. Loctite will hold the whole thing together with no problems if the fit is correct between the rod and the holes. This design should work well for the light loads that it will be subjected to on a watchmakers lathe.