Present perfect or past tense in the first part?

Are the following two sentences grammatically correct?

1. Mark has been injured while he was training.
2. While Mark was training, he has injured himself.

I am particularly interested in the correct tense of the verb “injure”.

In the first sentence, "while he was training" correctly modifies "injured", so the first sentence is correct in an appropriate context.

In the second sentence, "while he was training" attempts to modify present tense "he has injured himself", but that does not work.