Halts for all rejects, but might accept/loop otherwise?

Question

If a Turing machine halts for all rejects of L but might accept/loop otherwise, how is L's recognizability classified?

Recognizability	Decidability	${\langle L\rangle}$	${\langle \overline{L}\rangle}$
Turing-Unrecognizable	Undecidable	loop or reject	loop or reject
Turing-Recognizable	Undecidable	accept	reject or loop
Turing-Recognizable	Decidable	accept	reject
Co-Turing-Recognizable	Undecidable	reject or loop	accept
Turing-???	Undecidable	accept or loop	reject
Co-Turing-???	Undecidable	reject	accept or loop

Answer 1

To answer the question

If a Turing machine halts for all rejects of L but might accept/loop otherwise, how is L's recognizability classified?

$L$ is Turing-unrecognizable, but $\overline{L}$ is Turing-recognizable.

To see this, let $L=\overline{A_{TM}}$ and $\overline{L}=A_{TM}$, where $A_{TM} = \{(M,w)|\space M \space \text{is a TM that accepts} \space w\}$.

We can create a TM $M_L$ for $L$ that simulates the execution of the input $M$ on $w$. This TM will always halt and rejects when $M$ accepts but it is not guaranteed to always halt when $M$ does not accept. This is still a valid TM but not a recognizer. In Sipser it was proven that $L$ is unrecognizable but $\overline{L} $ is recognizable.

I think your table is a little misleading since it somehow hides the possibility of $L$ or its complement being TM-recognizable but not TM-decidable. Or maybe you overlooked the possibility that there are TM that are not recognizers. I am not sure what is the proper term to be used for them, though.