Well, let's start with that Samsung drive you have. It's fine for backups, but I would format it and wipe away the Samsung software. It may serve some purpose, but as you can see it got in the way of a really simple function you were trying to accomplish... Backup pictures and files and move them to another Mac.
Your Mac has an app called Time Machine built in. It is for backing up your files and such. As a built in solution, it generally works very well. The are other options such as CCC (Carbon Copy Cloner) and Super Duper which can both do what Time Machine does and more. They can clone your Mac's internal disk to another disk and if anything happens, you can boot from that disk as if nothing happened.
That said, you may just want to keep it simple. I have a disk dedicated to Time Machine and it's backups. I also have a second disk that I periodically backup my docs and photos to. I use a program called GetBackup Pro for that. It simply syncs the files I choose to the secondary backup disk. I do a one way sync so that it basically copies the changes I make to the source folder to the secondary backup. I use Time Machine if I need to go back in time for a file or if the system craps out and I have to start from scratch. CCC and SuperDuper give you a more "foolproof" way to backup with cloning, but I find them to be more than I need, and more technical to use.
What size is the internal drive on your Mac and what size is the T5 drive?