While I am by far no expert with Bluetooth, I think you need to set the hciconfig class.
man hciconfig states this for class:
class [class]
With no class, prints class of device. Otherwise, sets class of device to class. class is a 24-bit hex number describing the class of device, as specified in section 1.2 of the
Bluetooth Assigned Numers document.
You can find this at hciconfig (1) - Linux Man Pages
I found a list of classes at BlueTooth's Baseband page.
In Table 9: The Peripheral Major Class keyboard/pointing device field it uses this bit config
5 4 3 2 Minor Device Class bit no. of CoD
0 0 0 1 Joystick
0 0 1 0 Gamepad
And Minor Device Class field - Toy Major Class has this:
7 6 5 4 3 2 Minor Device Class bit no. of CoD
0 0 0 1 0 0 Controller
0 0 0 1 0 1 Game
If this helps, here is how I get BlueTooth information from a connected device:
pi@lucaspi:~ $ hcitool con
Connections:
> ACL **00:26:43:66:A8:74** handle 11 state 1 lm SLAVE
pi@lucaspi:~ $ hcitool info **00:26:43:66:A8:74**
Requesting information ...
BD Address: 00:26:43:66:A8:74
OUI Company: Alps Electric Co., Ltd (00-26-43)
Device Name: PLAYSTATION(R)3 Controller
LMP Version: 2.0 (0x3) LMP Subversion: 0x131f
Manufacturer: Cambridge Silicon Radio (10)
Features: 0xfc 0x07 0x82 0x7e 0x08 0x18 0x00 0x80
<encryption> <slot offset> <timing accuracy> <role switch>
<hold mode> <sniff mode> <park state> <RSSI> <channel quality>
<paging scheme> <broadcast encrypt> <EDR ACL 2 Mbps>
<EDR ACL 3 Mbps> <enhanced iscan> <interlaced iscan>
<interlaced pscan> <inquiry with RSSI> <AFH cap. slave>
<AFH cap. master> <AFH class. master> <extended features>
Try connecting your pi 0 to your other pi/debian system and see what info is different.
Lastly, maybe check out Bluetooth CoD: Bluetooth Class of Device/Class of Service Explained for information on CoD info. I will be reading this as well for one of my projects, so I might be able to provide more info.