While the exact extent of how "open" a certain's game world is tends to be influenced by several things, the crux of the definition lies in the core of the game itself - its goals, things you can do in it and the ways you can go through it. The more non-linear those ways are the more "open" the world is. And I'm not speaking about physical paths, but more about how varied your set of major destinations can be. That, in turn, depends on how much those goals, both mandatory and optional, are independent from each other.
To point at some examples, VVVVVV is a good example of an open world game. Outside of intro and finale, you are free to rescue your crew members in any order whatsoever. Now compare that to Blaster Master, or any IGA-made Castlevania. You go through some "hub" locations several times, but the order of "end-point" locations is always the same, enforced by needing to aquire new abilities from them to visit the next one.
Having optional goals (and I mean actual goals instead of just resource pit-stops) makes things a bit more complicated. The main storyline might be straight and linear, but if the game has enough side missions, optional questlines or just plain things to do to increase playtime at least twofold then your exact path through the game can still vary greatly. That's why Elder Scrolls and GTA series are seen as prime examples of open world games. But it's important that all this optional stuff stays independent from the main goals so that you can tackle it whenever you want. Exceptions can exist, but too many of them can diminish the "open-ness" of the world greatly.
Lastly, it's fine for the world to open up gradually, but the speed of opening up can influence how "open" the world feels. "Beef gates" and level restrictions fall under this as well. The downside of the game "opening up" too slowly is the gradual exhausting of the optional stuff. And what's the worth of an open world if it's already empty for you? JRPGs and games like Fallout often fall victim to this issue.
These are pretty much the cliff notes of how I see this question (due to me being too busy lazy to talk more about that). I might be wrong somewhere due to misremembering stuff, but this is pretty much my opinion on the matter.
(Also, "no loading screens" ROFLMAO. Even in "seamless" worlds there's a whole lot of loading and unloading going on. But that's another question entirely...)