As has been mentioned in the comments, although you have gathered together quite a few definitions and examples of usage, a common thread runs through them all in terms of what 'figure out' means.
In a broad sense of the word, to figure out something means to understand something clearly, however, what differentiates it from similar expressions such as to realise something or to find out something is the process involved in coming to that conclusion.
When we realise something or find out something, we are able to comprehend something that we couldn't comprehend before, but that may not have been our intention. Realisations can often happen by chance - something may spark a connection, and you then see it in a new light.
For example:
I just realised that there's an arrow hidden between the E and the
x in the 'FedEx' logo!
Similarly with 'finding out something' - we may have come by the information passively.
I just found out that my girlfriend was cheating on me.
Figuring something out, on the other hand, implies a conscious effort to study a matter or situation and actively search for the answer or understanding of it. Rather than by chance or happenstance, it is normally done with a specific goal in mind.
For example:
- We need to figure out what went wrong.
- It took them ages to figure out how to assemble the chest of drawers.
- When you've figured the answer out, put your hand up.
- They spent 20 minutes figuring out how to open the box.
There is a broad range of examples above, but all of these exhibit intent to come to a conclusion.
Being annoyed is different from being angry. If I am angry at someone, I might yell at them. If I am annoyed, I might just roll my eyes and shake my head. If I am greatly annoyed, I might exaggerate the eye-rolling and head-shaking, but it's still not the same as being angry.
Naturally this depends on context and personality. There are people who yell even when they are only annoyed. There are people who don't yell even when they are furious.
In addition, "annoyed" can be a diplomatic substitute for "anger". In a professional environment it is often inappropriate to be or act angry, so instead we use "annoyed" (or "upset") to downplay the severity of the emotion.
HR Rep: Ted, we've called this meeting because we were told you got a little annoyed with your boss the other day.
Ted: Yes, I was a little upset.
HR Rep: Well, your coworkers said you were screaming and running around the office, yelling out some very rude words to describe your boss.
Ted: OK, I guess I was more than a little upset.
For this reason we can only guess what "great annoyance" actually means. More context is needed to understand what actions resulted from the annoyance of his colleagues. Only then we can tell whether the author is downplaying a more serious emotion (like anger), or if it means the colleagues simply rolled their eyes and shook their heads frequently and intensely.
Best Answer
In this case, the phrase can be replaced with "that are not" and the sentence will still have the same meaning.
"Other than" is for exclusion. Whatever is after the phrase is not in the category. The example lists the new ways libraries can store information. Paper must obviously be excluded from the list because it is not one of these new methods.
Borrowing from BillJ's comment, we could also use "besides" or "apart from"
"Except" doesn't work for this case.
"Except" would be used if printed pages was a new storage method that wasn't being taken up by libraries. "Except" can be used as a substitute for "other than" in some situations.