I don't think you're creating the USB incorrectly, but if that were the case, I believe Ubuntu has its own official tool to create boot media that you can install using apt.

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Installation type Now, you can choose your preferences, such as the language and the installation - we recommend selecting Normal installation. Confirm your choices by clicking on Continue. Do not ...

If that doesn't work, then I really need to find a way to boot this machine and install Ubuntu without swapping the drives. I tried using Unetbootin to create a USB flash drive, but the Mac won't ...

The simplest involves downloading Ian Morrison’s “unofficial official” Ubuntu 14.04 disc image, writing it to a USB flash drive, and using that to install the operating system.