"Each theory has a different understanding of how countries like China, Japan and South Korea behave towards each other."
「[互]{たが}いの[理論]{りろん}はなぜのような[国]{くに}[中国]{ちゅうごく}や、[日本]{にほん}や、[韓国]{かんこく}を[始]{はじ}め、[互]{たが}いに[振]{ふ}る[舞]{ま}ることが[分]{わ}かることの[違]{ちが}っています。」
Vocabulary & Collocation:
1) How many theories are there in total? 「互いの理論」 would generally refer to "two theories" -- one of Person A and the other of Person B. If more than two, I would use 「[各]{かく}理論」 or 「それぞれの理論」.
2) We do not say 「なぜのような」 to mean anything; It is just not a good collocation. Besides, why use 「なぜ」 when there is no "why" in the original in the first place? The expression you need to use would be 「どのように」、「どう」, etc. as those are what "how" means.
3) If you used the structure 「～～や～～をはじめ」, it would need to be immediately followed by a noun phrase such as 「[多]{おお}くの[国々]{くにぐに} = "many countries"」. Because there is no such phrase in the original to begin with, you could not use 「～～をはじめ」 in your translation.
4) 「分かることの違っています」 makes little sense. I can easily tell you tried hard, though.
5) 振る舞う ＞ 振る舞る
" Perhaps I should treat that phrase 'different understanding' as a noun in Japanese?"
Exactly. It is a noun phrase in English as well.
My own TL attempt:
「それぞれの理論は、中国、日本、そして韓国といった国々がお互いに対してどのように振舞っているかについて、[異]{こと}なった[見解]{けんかい}を[示]{しめ}しています。」
I used the vocabulary and grammar you used as much as possible. Towards the end, I chose the words that you did not, but that is only because those would sound more natural to native speakers.
「異なった」= different,「見解」= view,「示す」= to show
