I listened to these guys yesterday. There are another one, where they use electric guitar. Very interesting talk they have. The youtube compression does kill the sound. But they do get into details that are way beyond my level, but still very interesting. Never heard about these people, but they do seem to know a little about music and stuff.
I have seen that string gauge video and came to a different conclusion than they did with the strings.
I agree they make a difference, but not a lot.
I personally think type of string makes a greater difference than gauge. So does the guitar.
Also, playing. I am definitely new enough that my playing is, shall we say, less than consistent. If I am careful, smooth and precise (with both hands) I get a fuller, warmer and sweeter sound. Carelessness makes lots of harsh overtones.
This is remarkably evident on nylon strings. Both the effect of string type and, tension and finesse.Guage, not so much.
For me, I like 12s (although of lower tension) because of how it feels. I dislike the razor wire feel of thinner strings. 13s might be fun, but I am a tension wimp from playing nylon and even 12s sometimes stick in the nut.
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The youtube compression makes it way more unclear. So many nuances get lost there. When listening to one who try to compare 11 to 13, like I have changed, the sound is way more different, than youtube can transfer.
I play somewhat like Nitsuj on his 7th day, maybe even worse… 
Not 100% sure about the nut problem. But as I understand, it is just to sand a little to widen the slices in the nut. That’s is what I would do without thinking much about it. Thus being extreme careful not to deepen the slice in the nut, as that would lower the string. But I prefer not to have to mess around with the nut. - I have slight problem with the 2nd string after the change, rest is sliding fine.
Just had a quick glance at nylon strings. Had to recheck few times. Had no clue it was like that and my logic says its wrong. Don’t want to dive into it now though. But that’s a big surprise for me. Very interesting. 
“”“”“Unlike steel strings, which tend to go up in a linear fashion from the high E to the low E (i.e. high E has the smallest diameter and then they go up progressively from there), classical strings diameters are non-linear.”“”“”“”“”"
I went to the shop today.
They had a look at the guitar and there should be no problem whatsoever using 13 strings. It could even take more force, but those thicker strings where only used for certain purpose and so on. Shop only sold from 9 - 13. They did though have single pack of 16-56. Mangan.
The sanding I had done with the saddle was really good. They looked, played, looked played like that several times.
The only thing they could come up with, was to play a little around with the truss rod. To tighten it just a tiny bit. ( Which actually was the opposite of what I thought they would say ). So very good I went there!!.
I changed those 3 packs of thinner strings to 13 - 56. So I have some extra strings and think they will last me long time during my journey with Justin as a teacher.
So beside playing a little with the truss rod, my setup is more or less as good as can be with this guitar.
Now full power on the teachings. 
I am happy as can be 
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