How to calculate cut-off frequency for T section high pass filter?

This is a high pass T filter: According to me, if input voltage is $V_{\text{in}}$, then voltage across the inductor $L$ should be $$V_x=\frac{j\omega L}{j\omega L + \frac{1}{2j\omega C}}V_{\text{in}}$$

Now, I think $V_{\text{out}} = V_x$ since we can only measure the EMF across the output terminals unless there's some load resistance places across those two output terminals.

In that case if we want the cut-off frequency, we'll need to set $|V_{\text{out}}/V_{\text{in}}|=\frac{1}{\sqrt{2}}$. I get $\omega_{\text{cut-off}} = \sqrt{\frac{1}{2(\sqrt{2}+1)LC}}$ from here. However, the actual cut-off (angular) frequency $\omega_{\text{cut-off}}$ (according to what we were told in class) should have been $\frac{1}{2\sqrt{LC}}$. I'm not sure why I'm not getting the same result.

Also, another question is: What is the use of the right hand side $2C$ capacitor of the T-section filter in determining the cut-off frequency? As long as no load is there, we should solely consider the voltage across the inductor $L$ to be the output voltage, isn't it? Or no?

With no load , the right hand 2C might as well be a short circuit and the 3 reactive parts only behave as a 2nd order high pass filter with high Q at the breakpoint.

With a resistive load the CR LPF adds a 3rd order HPF result.

( In theory with 0 ohm source and infinite load , the breakpoint has infinite Q , but in practice limited to Q = few hundred from ratings of components e.g. 100 is not hard, 1000 is not likely) With a load R this reduces by Z(f)/R at resonance.

What you see in the picture is the textbook representation of a 3rd order section, as Tony mentions in his answer, which does not include the terminating I/O impedances, which should exist in a real-life scenario. This means the transfer function becomes:

$$H(s)=\frac{Z_oLC^2s^3}{(Z_i+Z_o)LC^2s^3+(Z_iZ_oC+2L)Cs^2+(Z_i+Z_o)Cs+1}$$

The output voltage is not across the inductor, otherwise that would no longer be a 3rd order cell, so the output voltage is taken across the terminating impedance, after the second capacitor.The cutoff frequency, or pulsation, is simply the denominator's $s^3$ term, adjusted for its power:

$$\omega=\frac{1}{2\pi[(Z_i+Z_o)LC^2]^\frac{1}{3}}$$

And here's a quick verification with LTspice, where I used unity-based values, and I/O resitances instead of impedances, for simplicity: And the capacitors have the value 2C so that the transfer function becomes flat across the passband, otherwise, with only C, it becomes (linear amplitude, for better viewing): 