File Moving Problem within SSD Drive

I have two SSD drives and both of them have the same problem as compared to traditional hard disks.
Namely, when I move any file from one place to another with the computer mouse, the file itself does move, but an apparent "copy" or original version of this file stays in the point of origin. For example, say I move an .mp3 file from the computer desktop into a folder. The file will re-locate to the folder, but the same file will also remain on the computer desktop. I don't have this problem with traditional hard drives, just both SSD drives.

Note that it could be any file (mp3, jpg, MS Word, etc) but the result is always the same.
And the relocation could be from any two points within the SSD Drive.

One SSD drive is a Mac Mini and the other one is within a Mac Pro desktop computer.
Both computers are up to date with the OS X and both drives do not show any problems with the disk themselves. My goal is to move files around the SSD drive without a copy being produced during the file moving process.

Has anyone else had this kind of problem with SSD drives?
Thanks for any tips on how to fix this file moving problem from one location to another.

I have an iMac that got upgraded from a HDD to SSD and I have a Retina Macbook Pro that came with a SSD, on both of these machines the files copy/move as expected based on what I'm doing. Actions with files on the same drive cause a move to happen, actions with an external drive and internal drive cause a copy to happen.

I can force a copy to happen with actions on the internal device only by holding down the Option key. Unless, I'm mistaken this is the normal behavior of OS X. The fact that you have an SSD as opposed to a HDD shouldn't interfere with this action..

It is strange that you've managed to reproduce this on multiple machines..

I only mention SSD vs. HDD because I don't have this file moving problem with my HDD drives.
So that makes me think it's an SSD issue, especially when I have two SSD computers that have the same problem.

On the other hand, perhaps this is an OS X Finder issue.
My best "theory" at this point is that the Finder in El Capitan OS X within each SSD drive is responsible for copying and moving files at the same time (instead of just moving files).

If this is the case, perhaps there is a way to edit or delete this Finder command in Terminal.
I've also tested both SSD drives with Disk Warrior and it reports no problems with the drives themselves.

My goal is to move (and not copy) files from one location to another within the same SSD drive by dragging them with the computer mouse and not using any keyboard commands.
As far as I know, this should be a reasonable expectation with SSD drives.