There are different ways and styles of refereeing a tabletop RPG campaign, regardless of the system or setting being used.
"How to be a referee" is almost never backed by hard rules, just suggestions at most. For example, a PbtA game might tell you to "be a fan of the players," nothing bad mechanically is going to happen if the GM is actually antagonistic towards them. Early D&D--at least the Gygaxian interpretation of it--was very PCs vs. GM, but clearly that changed dramatically from table to table even within those early editions.
To me, this is all strange.
Without getting too metaphysical, a
game is a type of social activity that is characterised by the rules, principles and practices that govern how it is played. Change those, and you change the game.
Obviously, it does no good to be too prescriptive about these things. For instance, when I play backgammon it is for fun with family members, and we don't use the rules for doubling; whereas I imagine that serious backgammon players with money on the table
would use those rules - and yet I'm still happy to describe my game, and theirs, under the common label
backgammon.
Likewise, we still call the game
chess whether or not it is being played with a clock.
But just as we don't want to be too prescriptive, we don't want to be so loose that we lose all purchase on fundamental contrasts. Focusing very narrowly just on
what dice are rolled or
what bonuses are applied won't shed much light on how play is actually working in a RPG.
In a RPG, the most important thing that has to happen, for the game to work, is for the participants to all agree on (i) what actions are permissible for the players to declare for their PCs; and (ii) on
what happens next, in the fiction, after a player declares an action for their PC. The systems of a RPG are aimed towards providing answers to these questions. So a radical change in how the answers are generated means a change in system.
GMing a sandbox seems to generally imply two things:
First, it strongly suggests, or implies, a certain relatively broad range of permissible action declarations for the players. They are permitted to declare that their PCs
do things or
journey to places without the GM having provided a prior cue that the players are expected to have their PCs do
this thing or go to
this place.
This range of permissible action declarations is at least part of what people are getting at when they talk about the "freedom" that players enjoy in a sandbox; and when they contrast sandbox play with very highly curated GM-driven play.
A game having, or not having, these sorts of permissions is pretty significant. For instance, if I am trying to work out whether or not I will enjoy playing an offered RPG session, knowing what sorts of permissions I will enjoy in relation to action declarations is much more important than knowing whether basic action resolution will be d20 or d%.
The second thing implied by "GMing a sandbox" is adoption of a particular way (or perhaps one of a bundle of ways that are related in various ways) of determining
what happens next: rather than the GM determining that by reference to a script or pre-authored plot, the GM does so by some other process that is more open to "variability" and more sensitive to where the players are having their PCs go, and perhaps also to what they are having their PCs do.
Not every process other than
following a script will necessarily be sandbox-y, however. I already mentioned upthread that I think it would be misleading to describe Burning Wheel as a sandbox game; although it is a very non-script-y RPG. To count as a sandbox, I think the process has to make
journeying matter in some way - journeying has to have a distinctive sort of heft in resolution.
But there are different ways of establishing that sort of heft: the map-and-key, tracking-movement-on-a-map style of classic D&D; the progress-tracks-and-waypoints style of Ironsworn; the landmarks-on-a-map-create-toll style of Torchbearer 2e; no doubt others too.
If I am thinking of joining a RPG that describes itself as a sandbox, I would want to know what sort of method is being used to handle journeying, and to permit resolution without scripting. These are all crucial aspects of the system. If
@Hussar tells me that he is playing D&D - because he's using the D&D combat rules - but then I turn up and travel is being done via his adaptation of Ironsworn rather than map and key and movement rates, I'm going to get a bit of a surprise despite the trademark on Hussar's rulebooks!
Whether one characterises these rules as "hard" or not doesn't really matter to me, and I don't see why it would matter to anyone else either. A rulebook for a voluntary leisure activity doesn't exercise power in itself - all RPG rules get their purchase by being taken up and accepted by a game's participants. But the idea that there is no particular correlation between mechanics and other rules and principles isn't right at all: for instance, Apocalypse World's resolution system, which requires the GM to decide what happens next at many points in the resolution process, will break down if the GM ceases to be a fan of the players: the hardholder's steading will be destroyed, the gunlugger's guns all broken down or wanting for ammo, the hocus's followers all committing ritual suicide, etc. Or to pick a different example, Burning Wheel without its rules for framing and consequence just becomes a particularly brutal dice pool variant of Rolemaster or RuneQuest.
In the context of sandboxing, the map-and-key technique of classic D&D won't be very satisfactory if the GM is just making up the map on the spot (and I don't mean here
determining in a procedurally rigorous way, like Appendix B of Gygax's DMG, but literally just making up as the fancy takes them). Because how is
just making it up any different from the scripting that sandbox play was supposed to be an alternative to?
TL;DR - to talk about
system without addressing the basic points of
what action declarations by players are permissible and
how it is decided what happens next is to miss what is most important about the rules and principles that we use when RPGing.