I've come across these words many times over the years, but I realized that I don't have a proper conceptualization of the concrete difference between them. Can I ask for some sample sentences which highlight the nuances in these words?
What confuses you is, perhaps, English. Let's try a dictionary:
1 : a shortened form of a written word or phrase used in place of the whole word or phrase
2 : the act or result of abbreviating something : abridgment
The Japanese word that corresponds to definition 1 is 略語. Among that, shortened forms of names (usually proper nouns) are especially called 略称. What 省略 rightly corresponds to is definition 2 of abbreviation.
I can teach you what the difference between 略称 and 省略.
略称 means that you can make a short-cut ONLY for one name/group. So, for example, UN in Japanese is 国際連合, but you can do 略称 as 国連.
On the other hand, 省略 is used when you skip/omit something, for example, you can say that 長さの単位を省略する。, which means that you omit the unit of length.