I must admit I’ve never played Half-Life either. (The last 3D action game I bought was the original Tomb Raider, back in '96.) But watching the video, I got the distinct impression that that’s not the normal course of the story…
Actually… the differences are quite minuscule. The purpose of the intro is the make you feel like it’s your day job.
There are subtle differences like at the reception when the guy tells you about a system crash they had today etc. (and of course when the experiment happens shit goes down and the actual game starts)
Half-Life is my number one FPS of all time. When it came out I loved its setting and how to story is told without ever seeing yourself from the outside or ever speaking a word.
I loved the movement and game mechanics.
But what really made it so outstanding was all the mods this game (engine) made possible.
Everyone has heard of Counter-Strike, the most famous one.
The one above (Sven Co-op) made it possible to fight with others against AI monsters.
I played countless HL mods back then, single and multiplayer and tinkered around with scripts, mapping etc.
Have you tried Blake Stone? It’s on GOG and you can compile a very nice modern engine for it too. It’s Wolfenstein 3D meets Half-Life.
Don’t take that too seriously. I figure that you can’t really do Half-Life prior to the Build-engine/Quake-engine era. But for the Wolf3D engine it makes a good effort. Unfortunately gamepads aren’t supported, because playing Wolf3D is a blast with the Xbox One controller.