I'm repairing a small lamp table. It has a small 1' diameter round piece of polished stone that sits on top of a wood stand. There is a round piece of wood at the top of the stand that split in half. Half is on the stand and the other half is on the bottom of the stone top. I plan to just glue the pieces back together because the weight of the stone will keep it on the stand. Question is wood glue or construction adhesive?
For what you described, use wood glue.
If you are gluing wood to wood, then wood glue will suffice. You will want to keep it clamped in some manner until the glue sets.
If you are gluing wood to stone or stone to stone, then you're going to need a glue compatible with both of those materials.
Super glue or Gorilla glue are other options.
Gorilla glue expands and must be clamped well. Regular super-glue doesn't work well with porous surfaces. Use a type made for model airplane construction or a gel type on wood. Carpenter's glue may be old-hat but it's great when used as intended. "TiteBond" brand is my fave; TB 1 is standard waterproof wood glue, TB2 is fast-drying, and TB3 is slow drying. Those 3 will handle 90%+ of your wood glue needs.
Phil
Wood glue needs to be clamped well and should only be used to attach wood to wood. Construction adhesive will depend on the materials you stick together.