Baby safe wood finish

Hello. I’m a mother in search of an expert that can help me understand wood finishes. I purchase only solid wood toys and typically opt for unfinished wood products on which I seal periodically in beeswax. I’m currently hunting for a solid wood walker (not an easy task as most are plywood) finished as follows:

Wood Machinery & Supplies

Sliding table saws

In 2006 Martin presented the first sliding table saw worldwide with a saw blade that could be tilted to both sides. In March 2010, the company says it set new industry standards in terms of performance and flexibility with the T75 PreX. Today Martin has more than 1,700 machines with 2x46° technology have been sold.

How to get a matte finish?

I am trying to repair a piece of high end furniture (pic #1) - it has this really nice matte finish usually (pic #2). I have no idea how to get it to look like this! Any sort of anything makes it shiny. The pictures might not show it, but it is REALLY matte. Thank you!