This is a simple question.
What would a civilization on a flat world call the directions on a compass?
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I'm going to avoid Sir Terry's nomenclature, because it is so tied into the Discworld series I think it'd stand out like a sore thumb if anyone else used it:
Axle - towards the centre. Also could actually be the centre; the flat world equivalent of the Arctic - leading logically to:
Antaxle - the edge, and towards the edge.
Spinwards - obviously with the spin.
Backwards - against the spin.
Not really happy with "backwards" though. You could follow the Axle / Antaxle scheme and have Antispinwards, but I like that even less.
I was also thinking you could also use nautical terms if your world is ocean-heavy enough for sailing terms to have plausibly entered common usage:
Runwards - in the direction of spin; running is sailing with the wind, so implies it is the "easy" direction - the one that goes with the flow, as it were.
Beatwise / tackwise / tackwards - against the spin, again using the sailing terms for sailing into the wind - going against the flow.
I quite like "tackwards" as it implies "backwards" while sounding quite fantasy-esque.
Just realised there are two very obvious possible terms for the radial directions:
Out and In
I used this Wikipedia page to ensure I remembered my terms correctly; it's a treasure trove of nautical words!
If compasses work then the Bridge directions are fine. We did not have to discover the north pole to have a north. If the sun rises and sets in the same general places every day then east and west are knowable and the same as ours.
You've already answered your own question in the comments but I'll elaborate because I'm not sure you've understood them properly.
Assuming your planet is a spinning disk (elephants and turtle not required, what does the turtle stand on anyway*)
Towards the nearest point on the edge
Towards the centre of the disk, the hub of the spin.
In the direction of the spin. You could replace this with clockwise.
Against the direction of the spin.
This is an old English word meaning anticlockwise and implies the "wrong direction".
"On the Disc, if you enchant a needle it will point to the hub, where the magical field is strongest", perpendicular to the needle is the tangent to the Turnwise/Widdershins circle.
The Zero Meridian
This is key to navigation, the arbitrary point from which all others are measured. The zero meridian, as per the Greenwich Meridian lies on the line from the hub to the rim through the dominant sea trading city. Everything else is calculated relative to that position. Absolute Grid references can be given as degrees round the circle from that point (Turnwise or Widdershins) and miles from the hub.
It should be noted that navigation on open water is going to be really hard. If you're actually concerned about accuracy in such things you'll have to build a system such that it can be realistically done. Or just ignore it completely, this is probably the better bet.
*It's turtles all the way down