In more mature trees the bark is very different and is probably the easiest way to tell them apart...Hornbeam bark is grey, smooth and muscular (the bark in your first pic does look like regular hornbeam) and Hophornbeam bark is browner and thready for a lack of a better term. Your photo in post #6 is what makes me suspicious. The bark on young hophornbeam can look almost like birch and sometimes has a gold tint. It really doesn't matter a lot but it is nice to know what you have. Their care and treatment is very similar. Typically Hop has much larger leaves but they reduce well over time. In my area, neither young or old hornbeam hold leaves during winter as the norm...that might be different where you are...trees can vary. I believe they both make seeds in the fall and the seed structures both look similar to hops...hophornbeam seed clusters are longer.
