Is there use of $\pi/8$-DQPSK as shown in MATLAB's documentation?
I believe the correct name is pi/8 D8PSK. The ETSI TETRA Release 2 uses pi/8 D8PSK (digital trunked mobile standard for Professional Mobile Radio and Public Access Mobile Radio). The purpose of pi/4 DQPSK and pi/8 D8PSK is to have lower peak to average power ratios by not allowing the trajectories to go through the origin. Lower peak to average ratios means you can drive power amplifiers in the transmitter further into saturation, resulting in higher power efficiency (more transmit power from available DC power).   pi/8 provides for higher data rates at the expense of the more SNR that would be needed to get the same error rate performance.

More info on TETRA:

(see page 40 for details on pi/8 modulation which is identical to what the OP has shown):
https://www.etsi.org/deliver/etsi_tr/102500_102599/102580/01.01.01_60/tr_102580v010101p.pdf

Motorola TEDS:
https://www.motorolasolutions.com/en_xu/products/tetra/teds.html