There really isn’t a substantive difference. Technically both will work in your previous notebook entry, and “at the time” and “at that time” are interchangeable in every instance I can think of. Although, “that” can be a bit more specific in some instances. But, “at the time” sounds more natural. “At that time” is more for a newspaper story, a court document, police report, school book, etc. I would use “at the time” if in doubt. It will most likely always work and sound/read much better. An example where both are correct, but "at the time" sounds/reads more natural: “When I was ten years old, my father bought a digital satellite receiver. It was very expensive at the time.” (correct, sounds/reads natural) “When I was ten years old, my father bought a digital satellite receiver. It was very expensive at that time.” (correct, but sounds/reads "stiff") This is not to be mistaken with one sounding formal or informal. "At the time" is simply much, much more common and can be used for both formal and informal instances.

They are the same - two ways of saying the same thing. Examples: At that time, I was working in China as an English teacher. At the time, I was working in China as an English teacher.

(difference)