AHA! I'm afraid there is some mis-information in answers above, but you have found a good solution in the Seagate Expansion unit, OP.
USB2 ports can supply up to 0.5 A current at 5 VDC to connected devices. USB3 raised that limit to 0.9 A.
My understanding is that VERY few HDD units - even 2.5" low-power ones used in laptops - can run on 0.5 A. In fact, it you review the older "Portable Hard Drives" sold for USB2 systems you will find two dominant designs regarding power. One includes a separate power adapter that you must plug into the external drive to get it to work. That is what you have already in the 1 TB unit you mention. The other was a special power cable that came with many of those "Portable Drives" that had TWO USB2 connectors on its end. You had to plug BOTH of those into separate host USB2 ports to get enough power to run the unit. In fact, some people have found that getting one of those old cords can allow users of the new USB3 Portable Drives to use them (at USB2 speeds, since that's the cable they are using) with computers that have only USB2 ports, but you still need to use TWO ports.
The USB3 system solved the power limit problem for "Portable Hard Drives". The new ones on the market use 2.5" HDD's designed for Laptops with several features to minimize their power requirements, and CAN operate perfectly well within the 0.9 A limit. The Transcend unit you cited first is a good example of that. But typically when you connect them to a single USB2 port (no special double-headed cable) they start out looking like they are working, but never actually succeed in reading or writing anything. They just return errors to the computer because they malfunction with too little power. I have never seen any "USB3 Portable Hard Drive" that can work on a standard USB2 port with no additional power supplied. I consider it unfortunate that the promotional material for all these devices fails to tell you this. The tell you they are "USB2 compatible", but don't tell you you have to figure out how to provide adequate power to get it to work.
For someone like you, OP who has only one USB2 port available on the host computer, the only way to get a USB3 Portable Hard Drive to work is to arrange for more power. One way, as I said, is to scrounge up one of those double-headed old USB2 cables and use that IF your computer has two ports. Another is to do what you have done, OP - find a USB3 Portable Hard drive that includes its own power "brick" that you MUST connect when using a USB2 port. The third way is what ArtPog was getting at. Rather than buy a ready-to-use Portable Hard Drive, buy an external enclosure with a USB3 interface (and often with additional optional interfaces) that comes with its own "power brick" (since you need one anyway) and install in it a HDD of your choosing. If you get a slightly large one, you will not be restricted to using only the small (and expensive) low-power (and performance) 2.5" laptop HDD's inside. You could use a higher-performance 3.5" HDD, which FOR SURE will need more than 0.5 A and more than 0.9 A and have a more powerful "Portable Hard Drive" system with higher capacity at a similar price to the small "Portable" units.
We had to deal with exactly this problem. We bought a Transcend Portable Hard Drive that was sold as a USB3 unit (but we did not realize that). It could not work on a USB2 port, even though it sort of looked like it was trying. I found a different solution for use with a laptop that had only USB2 ports. I bought a powered USB3 Hub - one with several USB3 ports that came with its own "Power Brick" sufficient to power all the ports of the Hub. Then any USB2 or USB3 cable to the laptop's USB2 port provided the data connection between Hub (and its connected Transcend drive), while the power for the drive was provided by the Hub's own power source. It works perfectly that way, but at USB2 data transfer speeds, of course.