I am going to ditto what Spiker said a bit.
But first you need to know exactly, and I do mean exactly, where the lots lines are between the two houses. If that means hiring a surveyor so be it.
IF the lot line is where the good grass ends and the slope starts, then a retaining wall as Spiker suggests is what I was thinking was needed. However, I would not get the neighbor involved nor would I dig except to correctly installed the retaining wall block. You don't need a high retaining wall, just tall enough and with enough slope to move the water to the front or back of the house. Not sure which would be correct.
Building a retaining wall is not rocket science, I built one as the foundation for our barn, but it is hard work and you must level the blocks and keep them from moving. The block companies have very good instructions on how to do this and I bought my blocks at Home Depot. If your area has clay you could use clay instead of plastic or rubber as the water shedding layer though I would be inclined to use both. We had to use a specific clay to cover our septic field so that surface water would move on down hill.
In our city house, we had a similar problem with water run off from a neighbors house but the run off was just leaving a soggy mess. I dug a simple trench and filled it with sand, in hindsight I should have used gravel, cover the sand with fabric and then dirt. The trench was only about 18 inches wide and deep and maybe 20 feet long. I had to stop digging the trench due to tree roots. That little trench was enough to stop the wet yard problem. The soil was hard pan clay and I doubt much water sunk below the trench since the hard pan was like concrete. There were some homes in our neighborhood that were sitting in holes with water running in from four sides. How in the heck that was allowed is mind boggling. No basements but even with slabs, water flowing into a hole with a house is not a good thing.
I think in your case, I would put in a pipe on top of clay, if I could get it, AND plastic/EDM and them back fill with gravel and slope to where the water can drain best. Put geotextile fabric on top of the gravel. At that point you could add soil for grass or add maybe river rock or something that would look good. I would want to add something to mark the lot lines after all of this was over done.
Part of the problem is the gutter on the front of the house. That gutter looks to be handling a fair amount of roof thus water. The gutter should be tied into the system to move the water away from the foundation. Retaining wall blocks are buried so that at least the first course is below grade so a trench is needed. It might be a good idea to have a bit wider trench to allow a drain pipe to handle the other rain that falls into the area between the house and the retaining wall. The space between the retaining wall and house is going to make a big trench ditch so water needs to be moved out of there was well.
Try not to make the space so tight that working/replacing the heat pump will be difficult.
Retaining walls do not have to be straight, they can have curves which might look better than a straight wall.
Later,
Dan