So if I am trying to free up space by backing up pictures so I can delete them from my computer, but still have access through Time Machine backup how do I do this?

So I am trying to free up space on my computer by downloading pictures to the external hard drive, then deleting them off computer.

Ok.

To find them will I just always have to go back to the Time Machine backup where the pictures were backed up?

Only if you back them up with Time Machine. But if you're just trying to make space on your HD then you wouldn't do that. You can simply move the library to an external disk:

Move your Photos library to save space on your Mac - Apple Support

You'll note that Time Machine is not mentioned in that process at all. That said, you still need a back up - where a back up is a copy on other hardware such as a second external disk.

True.

But it's pretty small and light weight, quite smaller than my phone. I'm not pushing Crucial-- I'm just really impressed with how small and fast these new portable SSDs are, including the Envoy.

If you want to save the photos you delete, so can use them later, it will not be safe to have the only copy as a Time Machine backup. Time Machine is meant to keep a safety copy of your data, not the only copy.

Time Machine will not keep deleted items forever, it is a rolling backup. When Time Machine runs out of space, Time Machine will delete older backups of data that have been deleted from your Mac. You will lose the photos sooner or later. Also, you cannot open a Photos Library as a part of a Time Machine backup. You will have to restore the library from Time Machine, before you can access the backed up version. Save the deleted photos on an external drive and include this external drive in your Time Machine backup.

In line with what Yer_Man suggested look at this portable SSD: 250GB, 500GB or 1TB OWC Envoy Pro mini 

Then backup both the laptop and the Mini to Time Machine.

I have my system library on an external hard drive. I save that library to the cloud and can access my library on all of my devices. I have over 500gb of photos.

I just gotta say, OT, that a Crucial X8 1TB SSD is 50% faster (by my measure) and $50 cheaper on Amazon-- about the same price as the Envoy 250 GB.

That may be true but I was thinking about portability and size. The OWC whole be much easier to transport with the laptop than the Crucial model.