Wikipedia states the hierarchy of separation axioms as:
$$ \underset{\text{(Kolmogorov)}}{T_0} \impliedby \underset{\text{(Fréchet)}}{T_1} \impliedby \underset{\text{(Hausdorff)}}{T_2} \impliedby \underset{\text{(Urysohn)}}{T_{2½}} \impliedby \underset{\text{(Regular)}}{T_3} \impliedby \underset{\text{(Tychonoff)}}{T_{3½}} \impliedby \underset{\text{(Normal)}}{T_4} \impliedby \underset{\text{(Completely normal)}}{T_5} \impliedby \underset{\text{(Perfectly normal)}}{T_6} $$
Why doesn't anyone label another property as a "T"? I could add space properties like this:
$$ \underset{\text{(Kolmogorov)}}{T_0} \impliedby \underset{\text{(Fréchet)}}{T_1} \impliedby \underset{\text{(US)}}{T_{1¼}} \impliedby \underset{\text{(Weakly Hausdorff)}}{T_{1½}} \impliedby \underset{\text{(KC)}}{T_{1¾}} \impliedby \underset{\text{(Hausdorff)}}{T_2} \impliedby \underset{\text{(Urysohn)}}{T_{2½}} \impliedby \underset{\text{(Regular)}}{T_3} \impliedby \underset{\text{(Tychonoff)}}{T_{3½}} \impliedby \underset{\text{(Normal)}}{T_4} \impliedby \underset{\text{(Completely normal)}}{T_5} \impliedby \underset{\text{(Perfectly normal)}}{T_6} \impliedby \underset{\text{(Metrizable)}}{T_7} \impliedby \underset{\text{(Completely metrizable)}}{T_8} \impliedby \underset{\text{(Discrete)}}{T_\infty} $$
Is this acceptable? In particular, why doesn't metrizability qualify as a separation axiom?
Your hierarchy is correct, but the question why nobody has introduced $T_x$-names for the additional properties cannot be reasonably answered. It is a question of mathematical tradition, and you cannot exclude someone will suggest to introduce another $T_x$-name, but it is uncertain whether this will be accepted in the mathematical community.
Metrizability has relations to separation axioms, but in my opinion it does not add a new quality in separating points and closed sets. Being completely metrizable definitely has no connection to separation.