That sure looks like rust or iron, but if products like Iron Out or White Brite are not removing it, then I don't know. You might want to try posting this on the Laundry Room forum. Here is a link that might be useful: Laundry Room forum

Since water popping was not part of the process. Any sweat and spills will open the pores allowing the stain to be darker. They will need to spot fix, since this is an installer issue. The ring is from their buffer and last two images are either rag marks or hand prints.

I am thankful to come upon this site since our pool installer made us feel like this is something they never see and do not want to take any accountability. They told us we need to hire a power washing company. We live in SW Florida and recently had a pool installed and chose Silver Travertine for our decking. We do know travertine is high maintenance but was never warned about the staining that would occur immediately after install. It looks awful and the fact that this was a huge upgrade in our pool build we are regretting going with it. After just a few weeks after install on our covered lanai all of the rust stains and white powder started appearing. Also all of the silver travertine around the pool area in the sun always feels like it has a chalky powder residue on top of it and never feels clean. We paid extra to have cement poured first under the pavers. On top of the cement they put a sand base down before laying the travertine. They claim that this is the proper install. Here are photos of the stone under the covered lanai vs. in the sun around the pool.

Make up a thick paste made of ground chalk and peroxide, cover the stain, and leave it for a few days and see if that will remove the stain. Good luck