Yes , you need to upgrade clients on all servers.Twice cause upgrading DB from 9.1 to 10.1 directly is not possible.
And if you are on old runtime client scanario you'll need to switch into new global one - then you are doing upgrade just by executing db6_update_client script.
BTW why not to 10.5 in two steps:
9.1 to 9.7 and then to 10.5 ?