What is the relationship between an Interrupt Service Routine and a Device Driver that registers to an IRQ?

Question

I have recently be reading up on how to write Linux Device Drivers and came across a Device Driver which basically "handles" IRQ1 (a keyboard event). The code itself is shown below:

#include <linux/kernel.h>
#include <linux/module.h>
#include <linux/interrupt.h>
#include <asm/io.h>

/* This function services keyboard interrupts */
irq_handler_t irq_handler (int irq, void *dev_id, struct pt_regs *regs) {
  static unsigned char scancode;

  /* Read keyboard status */
  scancode = inb (0x60);

  if ((scancode == 0x01) || (scancode == 0x81)) {
    printk("You pressed Esc !\n");
  }

  return (irq_handler_t) IRQ_HANDLED;
}

/* Initialize the module and Register the IRQ handler */
static int __init keybrd_int_register(void) {
  int result;

  /* Request IRQ 1, the keyboard IRQ */    
  result = request_irq (
      1, 
      (irq_handler_t) irq_handler, 
      IRQF_SHARED, 
      "keyboard_stats_irq", 
      (void *)(irq_handler)
  );

  if (result) {
    printk(KERN_INFO "can't get shared interrupt for keyboard\n");
  }

  return result;
}

/* Remove the interrupt handler */
static void __exit keybrd_int_unregister(void) {
  free_irq(1, (void *)(irq_handler)); /* can't pass NULL, this is a shared interrupt handler! */
}

MODULE_LICENSE ("GPL");
module_init(keybrd_int_register);
module_exit(keybrd_int_unregister);

I would like to understand whether this Device Driver is considered an "Interrupt Service Routine"? Correct me if I'm wrong, but the flow of events is as follows:

1. Press key on Keyboard
2. Keyboard Triggers IRQ1
3. Operating System looks up ALL the Interrupt Service Routines "registered" to IRQ1 (which includes the Device Driver we wrote above) and triggers the code in these Interrupt Service Routines (including the code in Device Driver above).

Does this mean tat the Device Driver IS the Interrupt Service Routine?

A follow up question would be, how would you actually write a real Keyboard Driver (not one that just naively prints out some value)? How do you tell the Operating System that a particular key was pressed and that it should inform the current running application that this key was pressed? What does the C code look like for this?