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I am looking to use a figure for my thesis.

The figure itself is however (heavily) adapted from another paper.

I am fine with reaching out to get permissions from both authors, but I am unsure if I need to. Who owns those rights?

Additionally, I am unsure about how to write that permission was given. Right now, I have:

The diagram is reproduced from X, but is adapted from the work of Y. Permission from both publishers was given.

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Your intent and suggested wording is fine. You need to, at least, seek permission from the copyright owner of the figure you actually use. If that paper correctly cites the earlier version then you don't need to do so also, but it is nice to do.

Note that a figure in a paper isn't like a few words. The latter can be quoted under fair use (with citation), not requiring explicit permission. But a figure can be interpreted as "a complete work" independent of the fact that it appears within another work. Thus permission is normally required unless the figure is so mundane as to be a simple expression "common knowledge".

But you seem to be on the right track here.

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