Love the stories of TR1 and TR2. No clue who Vicky Arnold is but she did an absolutely outstanding job writing stories that were exactly the right complexity and depth for a videogame. None of the longwinded shit that you get in so many games nowadays.
Give the player an intro to hook them in (TR1: Lara is approached by Natla corporation to search for an artifiact of supposedly unknown nature in Peru. TR2: In a battle which took place in China centuries ago, an enchanted dagger allowed a general to transform into a dragon. Lara has heard of this legend and is on the hunt for the dagger)
Throw in a plot twist to send the player to the next location (TR1: Natla sends an assassin to kill Lara, prompting her to break into Natla's headquarters and discover more about the scion. TR2: Lara is fired upon by Bartoli's hitmen, and learns that they are also searching for the dagger)
Reveal more about the core plot, send the player to the next location (TR1: Lara combines the scion pieces and receives a vision of Atlantis. TR2: Lara is taken prisoner aboard Bartoli's oil rig and learns that, to access the dagger, she must race against the cult to recover the Seraph and enter the Tibetan monestary)
Final twist near the end (TR1: The scion reveals that Natla was an Atlantean ruler, imprisoned in stasis for unethical genetic experiments. TR2: Lara falls through the floor (lol) and lands in an expansive underground complex beneath the Great Wall, and sees Bartoli stab himself through the heart with the dagger. The pursuit leads through a strange realm made of jade where ancient Chinese warriors still exist)
Finale (TR1: Lara defeats Natla's half-formed ubermutant, destroys the scion, kills Natla in one-on-one combat, and escapes Atlantis seconds before it explodes. TR2: Lara kills Bartoli and retrieves the dagger, then escapes the collapsing complex just in time (and then fights the cult at her house).)
These are great stories for videogames because they give context for where you are and what you're doing at any given time, they offer clear goals, they lend themselves to many great and memorable set pieces (the Midas hand, Thor's hammer, the running shootout through Venice, the shipwreck which is partially upside down, the snowmobile chase, etc), and they can be told mostly through the environment and relayed to the player through exploration. Short FMVs cover the parts that can't realistically be put into gameplay - Lara escaping Natla's hitmen by throwing herself off the canyon, for example, or the plane crash in Tibet - and these are generally entertaining enough to act as a reward for finishing a level. Lara is wisely left as a mostly silent protagonist, speaking only in cutscenes. This gives her enough personality and presence that you know who you're meant to be playing as, but also makes sure that gameplay is never interrupted.
They feel similar to Half-Life to me - a relatively simple story that engages the player by actively placing them in it, and never halts the gameplay or removes the player's control to impart information because virtually everything you need to know can be relayed to you by just playing the game.