You know how when Deus Ex 3 first came out, it had boss fights that you couldn't get around, you were just forced to fight and kill a guy? And everyone was mad? And later the devs patched in other options, basically one for every style of play you might be into?
Well I've decided they were right the first time
I know this post is about Deus Ex but something I've always loved about Hitman is the way the tools and abilities the player has access to and the levels the action takes place in both feel fully formed and yet almost independent of each other. I never feel invited to approach a space in one of 5 or 6 specific, designed ways. Rather it feels like a space simply Exists, and I have all these tools and skills, and it's up to me to decide how these skills can be applied within this space to accomplish the goal I've been given.
More generally, I like when level design and quest design can create the feeling that you're just, like, in a place accomplishing a goal by any means necessary, rather than checking off a series of objectives. I don't just like "player freedom," I specifically like taking initiative, and that requires enough friction and uncertainty that there's actually some initiative to take, rather than simply having a game where I approach every challenge by doing whatever I specced into in hour one.
tl;dr: The actual distinction here is probably purely one of craft and skilled subterfuge but I find "player freedom" as in "4 or 5 predefined routes for every playstyle" kinda boring and "player initiative" as in "here's a location and a goal and a bunch of systems. make something happen with them" extremely compelling.