Almost correct.
A: No altitude info. You'd be seen on TCAS without any information and correct, not TA or RA even when you would eventually hit each other.
C (no TCAS): Altitude provided. Other aircraft's TCAS will indicate traffic, proximity traffic, and later on issue TA and RA if necessary as well. The TCAS equipped aircraft will be steered away from you. If you happen to apply your own evasive action - and it is your responsibility to do so - into the manouvering traffic you'll get close, but the TCAS on the other plane should be able to provide further corrective commands.
C (TCAS on): Further benefit is, that if both boxes issue RA (and that is normally the case) their commands-advisories will have had their sense co-ordinated. One up, the other one down.
Fd
(the un-real)