The guys quote was $3.15 a square foot without staining. It was another $.75 per sq foot with staining.

Three things:

1. The floor in your photo is definitely stained, regardless of the poly finish.

2. Oil based poly will add warmth to the look of a floor and darken it a bit, but not as dark as the floor in the photo from Houzz. Water base poly will be very clear, not add warmth and will darken the floors just a touch.

3. Oil poly will amber (yellow) over time. Water base poly will not amber over time, but the wood itself (oak) will yellow over time.

I would definitely get more quotes. A good HW installer will do a sample for you to approve. If you are going with a poly finish make sure your sample has stain AND poly on it for your approval.

When my very old red oak (at least I think that they are red oak) floors needed to be refinished, they were in bad shape with melted-on old carpet padding that needed to be sanded off. They were sanded down and refinished with oilbased poly only and had quite a lot of color then; after 13 years or so they have darkened a bit to somewhere between a prefinished Bruce butterscotch and gunstock although the shade is more golden than those samples. I will see if I can post a pic later. I have been told since that they had to have been restained but I am quite sure that he did not. Perhaps there was some residual color that persisted after sanding, although I do seem to recall seeing the floors after sanding and they were pretty natural.

Two things strike me right off. I question how your prospective floor finisher could say your old finish was "cheap Minwax". In my experience, most Minwax products are pretty decent. Second, the finish he's proposing would likely turn yellow with age and certainly WILL NOT give you the look in that picture.
The floor in the picture is definitely stained. If you were looking for a Minwax stain you'd want to look in the range of something like Early American or Provincial. In a Dura Seal, maybe Medium Brown or Chestnut. They have many mid-range brown colors that have varying degrees of red, brown and gold so insist on samples with stain and at least one coat of polyurethane.
I am in south Florida. Standards for refinishing are rather high so we charge $3.40 per square foot for natural color using oil-modified polyurethane, including using dust pickup equipment and an extra $0.35 per foot for staining. I have a legitimate business that's licensed and insured with someone to answer the phone, a showroom, a warehouse, a full time truck driver and mostly hourly employees so guys working out of their van can undercut me. Depending on the nature of the business your person has, his price my be fair or maybe it's on the high side. If he's charging that kind of price, ask for references and check him out on the internet. If he's proposing to use water-borne urethane, that could add to the price but make sure he's using a quality product. Not all products are created equal. Of course if you have a small area that could cost more per square foot.
The differences between doing stain and natural are taking a little more care sanding to remove sanding cross scratches that would be accentuated by the stain, a slightly higher cost of labor applying stain and maybe a bit more in materials for stain and wiping cloths.
I think you're wise to get some other opinions and prices.
Good luck!

"most Minwax products are pretty decent. "

That is giving Minwax way more credit than they deserve.

It is a bottom of the barrel low cost product.

The 'stains' are often a combination of dye and pigment and a PITA to get even over large areas.

They are not kidding when they say 'stir often during application.'

When I say "most Minwax products" I'm referring to their stains which are admittedly intended more for furniture use and their oil-modified polyurethane which I have used and had zero products with. Until Dura Seal came out with their quick dry stains in the Minwax colors we were still using them. It's hard to match colors like Jacobean or Dark Walnut otherwise.
Again, I still question how a supposed professional can walk in the door and say someone's old finish is probably "cheap Minwax" If you have spent any time at all in the flooring industry you know that there a lot of things sold out there that are far lower quality than Minwax.

I just posted a similar thread to this. I too was told by the refinishers that I spoke to that often times when folks refinish they do not stain the floors. However, drying time is the MOST important factor to us and I am wondering if any of you know, do they make fast drying oil based poly, or are the ultra quick drying products all water based?