Is LP distortion the record or the cartridge?

Dear @mglik  : Other than the alignment issue, that's is critical including that the Triplanar been mounted at the rigth distance from the TT spindle, your cartridge is alow compliance design and could be not very good tracker and its resonance frequency with your tonearm is around 12hz that could disturb a little the quality sound we are listening with. Your tonearm comes with a silicon damping tray that I don't know if you are using it: do you?

Now, how many playing  hours has your cartridge from new and which with which protractor made it its alignment set up?
Talking of the cartridge ( any ) what you are experienced looks like could be a cartridge suspension problem.

Btw, which phono stage was using before this " trouble "?

Regards and enjoy the MUSIC NOT DISTORTIONS,
R.

You wrote, "On dynamic peaks, even at fairly low volume, the sound breaks up and produces distortion."  On equipment that is at the level of your ensemble, those distortions should not be a bother, and they are not inherent to the vinyl medium.  In fact, elimination of those sorts of problems is one of the best justifications for spending big bucks on your vinyl rig.  Or, to re-phrase what edcyn wrote, "distortion and mistracking are [not] the bane of the vinyl universe".
I have no idea why you are having problems apart from the real possibility that your LPs have been abused by a previous owner, and there's not much you can do about that.  Before I would spend more bucks on equipment (like the Lyra Atlas), I would recommend that you purchase several known good LPs and first see whether you perceive they have the same issues when played on your system.  For example, buy some new high quality re-issues of the same music that is currently giving you problems on your used LPs.  The only other possible issue that is self-evident is the match of the compliance of your Shilabe (very low) with the effective mass of the Triplanar (in the low medium range).  The Shilabe might like a much higher effective mass tonearm.  Conversely, the Lyra Atlas might be a better match for the Triplanar.

@lewm , for certain. He can add some mass. Soundsmith makes a nice set of graduated cartridge screws. It certainly is not the Atma-Sphere overloading. I have a sneaking suspicion it is just worn records. 
The Lyra Atlas is a fine cartridge certainly in the upper echelon of cartridges. Not the best value but hey, if you have the money why not?
@mglik , what are you using for loudspeakers?

I think the culprit is worn vinyl.
I do have a lot of used records. Guess I will flag the worst ones.
I think the greatly increased performance of my new Atma MP-1 just picks up and amplifies the wear issues.
The good news is that the 180 gram remasters do seem to all lack this overt distortion. I know they are not as good as good as good first pressings or Better Records but they are very enjoyable.

@mglik , What speakers are you using?
There are certainly times when the reissues are better than the original particularly when it comes to older popular music. The quality of the pressings was not near that of classical music and in the late 50's and 60's recording engineers doing popular music were not as experienced. As an example the early masters of Zappa's first 4 or five discs were terrible. The remasters are fabulous. The reissue of Weather Report's Sweetnighter is significantly superior to the original. I would rather have a good clean pressing at 9/10ths sound than a scratchy one at 10/10ths.