Can badminton shoes used for jogging?

i never own a running shoes before since secondary school , i been wearing yonex shoes until now age 33 never try other brand shoes before , the latest shoes i owning now is SHB 03 Z mens

been wearing the badminton shoes for all training when school day

i just wonder is that very big different between running shoes vs badminton shoes for jogging?

So the last 2 week i start my jogging plan by wearing the shb03z men , it very comfortable for sure and never bring me problem , but i feel it very wasted because my jogging route is rough and i worry that my badminton shoes will be damage , so that i decide to buy a cheaper shoes for jogging and then finally i bought the asics gel kayano for around RM138 / 35USD and i sure that it’s a fake shoes

after 2 season of jogging , i feel it not as comfort as my SHB03z , the feeling is like day and night , it make my leg finger blemish

Now i have 2 option , i plan to buy the Yonex power Cushion Plus Eclipsion Z to replace my Shb03z men in court , and then take the shb03z for my jogging shoes

OR , buy a yonex running shoes for jogging and continue using the shb03z in court

So i really need ur guys opinion who have jogging + playing badminton , really appreciate if u could help me out

*I Will never buy or try a fake shoes from now seriously