"Click-Lock", "Engineered", "Engineered Click", and "Solid" hardwood... Is there a difference between "Engineered" and "Engineered-Click"?
Can you share the SKU on the "Engineered Click" option you're seeing? At that price it's likely HDF plus a wood veneer or thin layer on top. You should also be able to tell if you can look at a cross-section of a plank. HDF is likely better than MDF/plywood/softwood for the layers and may or may not be better than hardwood depending on who you ask, but it is definitely cheaper since it's a recycled product.

I was able to find the more expensive products you shared on the HD website, and they're real wood throughout (no HDF). Both are engineered in that they are in layers rather than one solid plank, even the click-lock.

The price difference between the two is just based on what thickness and installation style you want. The more expensive one per square foot is thicker (1/2" vs 3/8").