I mainly master to vinyl so I like to think of it as Lat/Vert but yes mid/side is a bit of a mis-nomenclature we are probably stuck with, just like "phase" being used in place of "polarity", though arguably not quite as misleading.
I use a simple thought experiment to get my head around the way it works.
M contains information from both L & R, and so does S. We know M = L+R, but what about S? Well I like to think of there being 2 versions of S... "SL" and "SR":
SL = M/2 - L
SR = M/2 - R
The difference between SL and SR is just a question of polarity. If we give L and R some arbitrary values, say L = 9 and R = 7, we can then give values to M and S (16 and 2).
So M/2 = 8 and SL = -1 and SR = 1. The two numbers will always be the same, but one will be negative and the other positive.
If we then change the value of S before decoding it back to L & R, say by adding 2 to give S = 4. Then effectively we have added 1 to L and taken away 1 from R, thus making the difference between the two greater, even though the sum is still the same:
M/2 + S/2 = 10 = L
M/2 - S/2 = 6 = R
That's why boosting S widens the signal, and reducing S narrows it. Of course being a wave, S will always have a range
of positive and negative values associated with it, but the effect is still the same. Its either adding to L and taking away from R, or vice versa depending on whether its value is positive or negative.
Not sure if that helps anyone to think of it that way, but it helped me! Incidentally, it is possible to set up a type of matrix whereby you can control SL and SR independently. Its actually a lot more simple than the standard M/S matrix, but I guess its not that popular as it requires an extra channel.