Why is it better to have a higher M2 and Sharpe ratio with a lower treynor measure?

CFA Level III
#1

A better ranking (higher score) for M2 and SR, but a lower ranking using Treynor vis a vis some other portfolio, indicates more diversification and less unsystematic risk. Can anyone explain this, please?

Some assumptions/vocabulary (correct if I’m wrong):

Standard Deviation = measure of total risk

Beta = measure of systematic risk only

#2

The alternative implies they’re not well diversified

#3

M2 and SR being higher indicates relatively low total risk.

Treynor being lower indicates relatively high systematic risk.

Therefore, the unsystematic risk is relatively lower and diversification is higher.