No, because spells whose area of effect is a line, by definition radiate in one direction only from their point of origin.
Wall of Fire is an Area of Effect spell, defined as spells which "cover an area, allowing them to affect multiple creatures at once" (see Spellcasting: Areas of Effect, and thanks to WakiNadiVellir for suggesting this approach to answering the question).
As the Areas of Effect rules explain,
A spell's description specifies its area of effect, which typically has one of five different shapes: cone, cube, cylinder, line, or sphere. Every area of effect has a point of origin, a location from which the spell's energy erupts. The rules for each shape specify how you position its point of origin. Typically, a point of origin is a point in space, but some spells have an area whose origin is a creature or an object. A spell's effect expands in straight lines from the point of origin.
Note that every AoE spell has a point of origin, so Wall of Fire must as well. Also note that while AoE spells are typically one of the five shapes and typically have a point of origin that is a point in space, it is the "spell's description" which specifies these, so Wall of Fire might be atypical, so long as its area of effect is described.
With these base rules in mind, we can now evaluate the the description for Wall of Fire
You can make the wall up to 60 feet long, 20 feet high, and 1 foot thick, or a ringed wall up to 20 feet in diameter, 20 feet high, and 1 foot thick. The wall is opaque and lasts for the duration. When the wall appears, each creature within its area must make a Dexterity saving throw. On a failed save, a creature takes 5d8 fire damage, or half as much damage on a successful save. One side of the wall, selected by you when you cast this spell, deals 5d8 fire damage to each creature that ends its turn within 10 feet of that side or inside the wall. A creature takes the same damage when it enters the wall for the first time on a turn or ends its turn there. The other side of the wall deals no damage.
With that description, what is the area of effect of a wall of fire? That depends on the effect, for the wall actually has two effects and their areas differ!
The first effect is visual; "the wall is opaque". This can be cast with a 'typical' line shape; "60 feet long, 20 feet high, and 1 foot thick" or an atypical ring shape, which is not one of the five shapes mentioned in the PHB but which is described by the spell; "a ringed wall up to 20 feet in diameter, 20 feet high, and 1 foot thick."
The second effect is damaging; "a creature takes 5d8 fire damage". The area of this effect overlaps with the visual effect but is not the same as it; in particular it extends beyond the wall itself; "One side of the wall, selected by you when you cast this spell, deals 5d8 fire damage to each creature that ends its turn within 10 feet of that side or inside the wall...The other side of the wall deals no damage."
This description best conforms to that of a line. A spell with an area of effect in the shape of a line functions as follows:
A line extends from its point of origin in a straight path up to its length and covers an area defined by its width.
In this case, the point of origin of the effect is not a literal point is space, but rather "one side of the wall, selected when you cast the spell". The 'point' of origin is in fact a line - the base of the damaging side of the wall (or in three dimensions, it is an area, the outer surface of the damaging side of the wall). From this origin, the damage radiates outward ten feet - this is the length of the area of effect. It covers an area as wide as the wall is long - this is the width of the area of effect. The damage does not extend to the other side of the wall precisely because the area of effect is a line, where the spell radiates in one direction only from the point of origin. This can be compared with another spell whose area of effect is a line, for example lightning bolt, which "blasts out from you in a [single] direction you choose". It is initially confusing that Wall of Fire's description says "The other side of the wall deals no damage", and not a more straightforward 'Creatures on the other side of the wall do not take damage from it'. However, by specifying that only one side of the wall deals damage, the spell is actually alerting us to the fact that its area of effect is a line, which we would not otherwise be able to determine.
Thus, when the OP says "After all, if I'm 9 feet South of the wall, I'm still standing within 10 feet of the North side of the wall..." this is true, but what it lacks is the understanding that the damaging effect radiates only to the North of the wall, since that is the nature of a line.
For a ringed wall, the point of origin is still the damaging side of the wall (which is now a curved line or area), the length of the area of effect is still the straight line radiating ten feet from the damaging side, and the width is still the (now curved) length of the wall.