I would think that the sound would be absorbed better by a drywall ceiling than by a wood ceiling, although I doubt the difference is that much.

Fabric would be much better for absorbing sound. Are there any draperies or rugs in the dining room??

No discernible difference between drywall and wood. You need soft surfaces: carpets, curtains, artwork. Even upholstered walls.

YEP

A slatted wood ceiling with gaps in between the slats would help. There are panels that can be installed or it can be done board by board.

Wood surfaces are pretty dense and should reflect sound as readily as most other solid surfaces.

Both are hard surfaces and not great at absorbing sound. Fabrics do this better than hard surfaces. Here is a link to an acoustics company that may be able to provide better guidance and wider choices.

https://www.acousticfields.com/free-room-analysis/

Thank you very much for all of the comments. Based on most of the opinions , we will try adding more fabric , wall decor, drapery, etc. before investing in an expensive install of a wood ceiling. If it does come to that we will look at the more acoustic sounding slated wood option or perhaps try it on a wall. Thank you again everyone. Your comments were a great help. I appreciate the time you spent in posted your opinions. I will post the results after we try various options in order to help someone else.

If I had to guess, as you didn't post a picture?

You've a hardwood floor, No area rug, a big window, it has a wood table, or yikes a glass top table , wood chairs with no upholstery. Now add a half dozen people, dishes , etc.

Result ? Din, and dinner : )

Upholstered chairs

Rug sized to fit all the dining table and chairs

Curtain panels.

Noisy at any height ?! right here

Sound absorbing.......soft chatter : ) and prettier too, below

( adjust to you and yours)