A Monster's Expedition had a particularly memorable way of doing this. After the first few puzzles show the basic mechanics, a situation is set up with impeccable timing where you end up taking an action you expect to do one thing to solve the current puzzle, but when the action is taken, it ends up revealing another mechanic which makes a dramatic change in how you perceive the game world. The game knows this is a big moment, and the revealed mechanic causes a bit of gameplay downtime, and it choses that moment to do a title drop.
I watched a let's play for it. Wow! That is an excellent sequence!
Megaman X is still my go-to. Recently, Doom 2016 did a pretty awesome job.
Megaman X is a masterclass on how to start a game. I like Capcom's habit of showing the final boss on the first level. They did it with Devil May Cry 5 as well. I love how Doom just starts with virtually no cut-scene.
Final Fantasy 7 (the original, haven't played the remake) is my go to example of a strong intro, not least because it sticks out in a genre that is kinda notorious for slow and overlong intros.
FF7's intro is fantastic. I love how the cut-scene literally zooms out as if to say "Here is the interesting game world" then zooms into the main characters doing an action set piece as if to say "These are the people you need to care about".
It's also a great example of a story that has an Act 1 that isn't boring.
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