Thank you for your response. Do you think Windex would work? I don't have any issues going out and buying something else - I just have Windex on hand.

One thing I should mention, though, is that I rubbed a damp sponge (the green side of dish sponges) with slightly soapy water on a small part of the marked floor and after a bit of a scrub, I was able to clear out some of the marks. It also looked like there was some dirt coming off - It looked more like eraser dust. That small part of the floor did clean up completely, but I was afraid I was removing the urethane coating and stopped, although, it may have just been this layer you speak off with dirt mixed in.

Generally these polishes can be removed with an ammoniated or non-ammoniated stripper. Ammonia can damage wood flooring (like pet urine stains) but with ammoniated Windex you should be able to determine whether there's a finish that could be stripped off. Try it and report back. If it does work I'll give you some tips for overall removal.

Just tried it and it worked. So it almost seems like the polish layer had gotten dirty and the Windex removed the layer. The floor below actually still has a sheen and no marks. This is great!

At this point you can buy some non-ammoniated stripper which is generally sold in large quantities or buy something like Armstrong New Beginning that you might be able to buy locally. My local Home Depot has it in stock. IT CAN DAMAGE wood floor so use it sparingly, doing small sections at a time, by hand. It will certainly work at full strength but might try diluting it with water first maybe starting at a 10 water: 1 stripper ratio and adding stripper until the effectiveness increases. Use water sparingly to rinse and remove dissolved polish. Another option would be to find and purchase Bona Polish Remover, which I would assume may be less likely to damage wood flooring since it's formulated to use on wood. I would still use it sparingly.

Thank you so much for all the advice. I appreciate it very much.

I was reading up on Armstrong New Beginning and it doesn't seem like it's safe for hardwood floors. While I have definitely taken into consideration your suggestion of using it sparingly, being the overly cautious and paranoid person that I am, I'm hesitant to use it *just in case* I mess the floors up.

I couldn't find Bona Floor Polish Remover anywhere close to me and in very few places online as well. I did, however, find a million negative reviews of the Bona Hardwood Polish, which I have somehow convinced myself into thinking is what is on my floor.

Do you think water+vinegar may help in this case? Not sure if using Windex for the rest of the floor is a good option either, but the results were just brilliant. It's probably overkill, but I'll post pictures of the small area I cleaned tomorrow. I'm just excited I don't have to spend a bunch in screening and refinishing the floors.

You can try vinegar and water but I doubt it will work. My own workmen have successfully used both of the products I mentioned. You're removing something that shouldn't have been used on a wood floor.
https://www.amazon.com/Bona-Polish-Remover-32-fl-oz/dp/B00550C396/ref=sr_1_1?ie=UTF8&qid=1500895449&sr=8-1&keywords=bona+polish+remover

Thank very much. The Bona was out of stock when I was looking for it over the weekend. I used Armstrong New Beginning, highly diluted at first and then increased concentration a little at a time. I'm at around 1 cup to a gallon right now.

On another part of the floor, I tried using a little diluted Ammonia. This seemed to work as well, but the Armstrong works better. Regardless of what I use, the floor needs a scrub to get that layer out.

I have around 800 sqft of wood flooring and I've been able to cover just 150 because of having to get down and scrub. Posted is a before and after image. The difference is tremendous. I just don't think I'll have the energy to scrub the entire floor myself.

Before and After:

This is a picture of the same area I had attached to my first post ^^

Wow! That's amazing. Try a 2 cup: 1 gallon of water in a discrete area and see if the stripper will do more of the work so you don't have to do so much scrubbing. You could even try a small area using a very thin layer at full strength.

Polishes are NASTY! That's why I hate them!!! It would be much easier if people didn't mess with polishes! It keeps things so much simpler.

The scrubbed area looks EXCELLENT! Sorry to hear you have another 650sf of this...this is precisely why grandma was happy to kick her "polisher" to the curb! A professional hardwood floor refinisher could do this with by using a screening pad and a buffer (and the mixture you created). They could probably get all of it off in a few hours. It would probably cost $0.50 - $1/sf or so. Depends on how much s/he likes you:-)

JFCI, Will try increasing the concentration.

CFI, I asked around - some went up as high as $1.25/sqft. One of them was more reasonable at $0.75/sqft, but after speaking with him I realized that I'd have to move all my furniture out of the way. My floor plan is a little weird - the main living area is split up into two halves by the kitchen in the middle. One side of the kitchen forms the living/dining/great room and the other my home office making this a two day job because of moving around the furniture. I will call upon the pros in the future for a complete floor facelift - right now, I'm just going to try this by hand so that my floors don't look like they've been pulled from a warehouse.

I'm considering getting a handheld polisher like this one - http://www.homedepot.com/p/Ryobi-18-Volt-ONE-10-in-Orbital-Buffer-Tool-Only-P435/205975768 to help with removing the sludge. This will also give me the flexibility to work around my schedule/furniture. I couldn't find any mildly abrasive pads, but I reckon the microfiber/cloth pads that come with the tool should suffice at high enough speeds.

Will update once I get out and but this thing! Thank you both for all the advice.

I hope the rest turns out as well as the area already done.