How can I know the type of filter from its cutoff frequency?
Types of filters, such as LPF, HPF, BPF or BSF, are not described by their cutoff frequencies $f_c$ but through their passband or stopband frequency $\omega$ region. 

For example, in your case of the filter $H(z) = 1 / (1 - 0.9 z^{-1})$, the passband is in the low frequencies (where the magnitude of the frequency response $H(e^{j\omega})$ is greater than $1/\sqrt{2}$. And by your definition that region corresponds to $\omega \in (-f_c , f_c)$ which indicates a lowpass range of frequencies.

Other filters like HPF and BPF follow in a similar manner. Note that specific filters such as comb or notch types are defined in a different style than this.