Unfortunately the word controller is overburdened in the MIDI lexicon and probably the most overused word in the world of MIDI.
It can refer to three different things- products, physical controls and messages.
People can say MIDI Controller and they mean a product like a IK MultiMedia iRig Keys I/O 25 Controller Keyboard.
They might say " I'm using The Roland A88 MK2 as my MIDI Controller".
But the word Controller is also used to refer to physical controls like a Modulation Wheel, a Pitch Bend wheel, a Sustain Pedal, or a Breath Controller (yes, there is that word again).
The word Controller is also used to describe the MIDI messages that are sent. So you could say "I'm sending Controller #74 to control Filter Cutoff'.
In fact, there are multiple types of MIDI messages that are sometimes referred to as "Controllers":
To make things a bit more convoluted, the MIDI 1.0 specification contains certain MIDI Messages that are named after physical controls specifically-
Decimal Hex Function
1 0x01 Modulation Wheel or Lever
2 0x02 Breath Controller
4 0x04 Foot Controller
11 0x0B Expression Controller
64 0x40 Damper Pedal on/off (Sustain)
66 0x42 Sostenuto On/Off
67 0x43 Soft Pedal On/Off
But these are MIDI Control Change (CC) messages, not the actual physical controllers themselves.
However most products hardwire the Mod Wheel to CC#1 and set the factory default of Damper to be assigned to CC#64, etc.
Also on most MIDI products you can set your physical controller Mod Wheel to send different CC messages (for example Control Change #2 Breath Controller or Control Change #11 Expression).
MOD WHEEL is a physical controller that always generates a specific message cc001 Modulation Wheel. cc001 (Control Change) can be applied to most any function, it does not have a fixed function. It is most often used to apply Modulation depth to pitch (vibrato) but that must be assigned to the wheel on a per program basis.by Yamaha Product Specialist Phil Clendennin ( AKA Bad Mister)
So a MIDI Controller has a MIDI Controller that sends a MIDI Controller! Or translated into a sentence that makes more sense-
A IK MultiMedia iRig Keys I/O 25 has a Mod Wheel that sends Control Change (CC) #11 Expression.