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The medium sized one looks like a good design; I did a similar one for a spice rack.
How many bottles, and how how heavy they are, and how often they are moved will influence the wood to use. For occasional home use of a few bottles, 1/4" plywood would be good. For a busy public bar, you would want something much sturdier.
I would glue AND screw it together. Use wood glue, not regular white/PVC/tacky glue, which are too flexible for this type of project.
And welcome to Lettuce Craft!