Water is a powerful force. If you re-grade, where will the water go? I would work with nature in your landscaping, not against it.

That's kind of what I was thinking. You're only showing a snippet with the photo, but if we saw the whole space would we conclude that the drainage swale was intentionally created in order to handle the water that must go somewhere? You may make changes but you'd better make sure ahead of time that they will work. Also, is grass not already there because of too much shade? The photo tells little of the existing conditions.

Dry stream bed would be awesome. There are plant choices that would do fine with wet conditions alternating with dry conditions. Can't say exactly what those might be since we don't know where you are geographically, what kind of climate. I think I see maples, so I assume you are somewhere in the temperate zones and not the tropics/subtropics.

Just an FYI, if you do think you want to pursue leveling/grading, better check first with your municipality to make sure it's ok. Where I live, a lot of people get in trouble by doing the work before getting a permit to change the drainage patterns or do work on wetlands, then have to pay to undo what they did.

Here is another view of the yard. We get all the neighbors run off. Water moves from the mid left of photo toward back center. It works great with the exception of the water just stands stagnant. I wish that it collected on the opposite side of the path or I could feed toward the back corner (top right in photo). I had thought about a dry stream bed, but my husband really wants a little bit more grass in this area.

Would there be a way for me to build up most of it, but leave one area lower, maybe north near the base of tree and create a dry bed that extends beneath hammock and to back of the property? That is where the actual maintenance ditch lies, and you can see how the land starts to slope downward.

Final look..viewed from west side and looking to east. You can see the dirt boundary in the yard where the grass stops. This is where the water collects. We are in central Indiana, btw.