This is how my professor is finding the frequency response of an LTI system when given the impulse response. Is this wrong?
Your professor is right, and you're almost right too. The filter is clearly an FIR filter, but because its frequency response can be expressed as a geometric series, a recursive implementation is possible.
If you write the transfer function as a rational function you get
$$H(z)=2\frac{1-z^{-12}}{1+z^{-2}}\tag{1}$$
which is almost the same as you got, apart from a constant factor.
Note that there is a pole-zero cancellation in $(1)$, so the filter is FIR with impulse response
h = [2  0 -2  0  2  0 -2  0  2  0 -2];
So apart from rounding errors both

b = 2*[1,zeros(1,11),-1];
a = [1,0,1];
[H1,w] = freqz(b,a,1024);

and

[H2,w] = freqz(h,1,1024);

must give the same result, which turns out to be true. On my machine I get

max(abs(H1-H2))
ans =    7.9623e-14

Note that the phase is ambiguous, so the phase values are only identical up to an additive multiple of $2\pi$.