The reason I said you should be only talking about enzyme concentration is because that's what you said at the top of your post.
If you have a constant substrate amount - somehow the reaction occurs but it replenishes itself. Say there is 100 substrate molecules. If you put in enough enzymes so that 100 reactions are always occurring at any given moment, then increasing the amount of enzymes beyond that will not increase reaction rate. It doesn't matter if you double the amount of enzymes, there are no more substrates for them to react with at any given time. So at this point substrate is the limiting factor - increasing enzyme concentration will not increase reaction rate because it is no longer limiting it (this will look like a plateau on a graph. X axis time, Y axis reaction rate.)
If you have a set number of enzymes and you increase the amount of substrate to the point where at any given time there are no enzymes available to catalyse a reaction then the enzyme concentration is the limiting factor. When this happens the enzymes are 'saturated' increasing the amount of enzymes will increase the rate of reaction until something else becomes the limiting factor.
increasing enzyme concentration has a diminishing impact on reaction rate is because substrate concentration is decreasing.
This is not entirely correct. It relates to what I said above. As the amount of substrate decreases, the amount of enzymes needed also decreases. Increasing enzymes above what is needed will always not increase reaction rate, the amount of substrate just changes the amount of enzymes that are needed.