Delay and Stablity in Negative Feedback Systems: Confusion
In your second example (2nd-order system with feedback) the phase of the loop gain function will approach the critical value of -180 deg for infinite frequencies only. Tha means: The phase shift will never reach -180deg at a fixed frequency - and the systenm will be stable. 

In your first example (first order with fixed delay block) the loop gain phase will not be limited to any fixed value. Instead, the phase will rise with the frequency without any limitation. Hence, if the loop gain is larger (smaller) than 0 dB at a total phase of -180 deg, the closed loop system will be unstable (stable).