Saying Emma's "signature is all over the bark" is like saying Jean's signature is all around
them. However, I did notice a hint of it around the branches. Nevertheless, when the lightning strikes the tree, it is not contorted nor is Emma's signature anywhere to be seen. Also, let's just say you are correct, and Emma actually did manage to do
something in that scene, it is clear she was saving herself as much as she was a caught-off-guard Jean. That is not the same as what Jean did for her on the previous pages. Emma was trapped and helpless; Jean was not. (See below regarding the tree not being contorted and showing no signs of Emma's power signature.)
The tree:
I will leave this debate at this: The scene can be open to interpretation.
I will allow you to use the Phoenix card. And yet, Jean, not bonded with even a shard of it, still yanked the Phoenix Force out of Emma with ease on two occasions...
(Admittedly, this is fun but crazy, too.)