I would like to inquire about the difference in meaning if I want to translate these two sentence into English:
Ich stehe zu dir
Ich stehe hinter dir
Could both sentences mean "I'm standing with you"? Or do they have different meanings?
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Both German sentences can (and often will) be used with the meaning of supporting someone (I've never heard the English I'm standing with you, so I'm not 100% sure if this is the correct translation).
They are not exactly the same, both can have other meanings as well.
Ich stehe hinter dir can also be meant in the literal way as "I'm standing behind you".
Ich stehe zu dir can also be used when another person (probably when you are not around) tells me that you are not good for me but I say that I'm not going to end our friendship because of that person's advice.
Although @Volker Landgraf did a good job explaining the difference with these terms I feel like there are a few suttle things that can be added here.
"Ich stehe zu dir" is more often than not used to imply a positive relation between the parties, like friends or intimate partners, meaning something like "I will be there at your side however bad your decisions may be".
"Ich stehe hinter dir" often implies a leading role of the addressed person. If A says to B "Ich stehe hinter dir", then A is saying that B does something and A has a supporting role in that matter. This phrase is often used when talking politics, for example when you are talking about the support of a political party for the idea proposed by a specific politician. "Die Partei steht geschlossen hinter dieser Entscheidung" "The party as a whole supports that decision."
I hope this explaination further improved the distinction between those two phrases.