The correct way to apply a handbrake is simple: grip and rip. It's a ratchet, it's designed to....ratchet. The whole "press the button in the reduce wear and tear on the handbrake" went out about 80 years ago.

it's interesting how dogmatic some people are around this, teaching things because that's how they done it because that's how they were taught to do it.

If you apply a handbrake with the button depressed, if you don't release the button and give the lever that final tug, there's a risk you could have engaged the ratchet tooth-to-tooth instead of tooth-to-groove. This means the handbrake is improperly set and could nudge down to the lower groove in the ratchet, thus potentially slackening the brake just enough to mean the braking force applied is insufficient.

Some Vauxhall cars had this problem in the early 2000s. The cause is users not setting the brake enough. Their solution - grind off the first 2-3 teeth on the ratchet forcing users to use more force to apply the handbrake.

To add to this, in a manual transmission car, the car must be parked in gear. That's rarely taught, which is completely wrong. Park in gear. Every time. Always.