Definition of Ideal Low pass filter (Time Continous)
An ideal low-pass filter has frequency response $H(f)$ equal to the brickwall function: $$H(f)=\mathbb I[|f|&lt;f_c]$$ where $\mathbb I[s]$ is the indicator function, which is equal to 1 if the statement $s$ is true and 0 otherwise, and $f_c$ is the cutoff frequency. Since this frequency response is real, its phase is zero.

This ideal filter is not stable and non-causal. It also has zero delay; it corresponds to $t_0=0$ in the first definition you list above. Now consider the case where we want an ideal LPF, but with some delay $t_0&gt;0$. If the filter input is a sinusoid of frequency $f_0&lt;f_c$, then the filter output is  delayed a time $$-\frac{-2\pi f_0t_0}{2\pi f_0}=t_0.$$

In other words, the first filter in your question is ideal with a delay $t_0$ that you can measure or specify; the second filter is ideal with delay equal to 0. As pointed out in another answer, this helps to make the filter "more causal" -- it allows you to implement a filter that approximates the ideal.

See here for another explanation of phase and delay.