I have analyzed the two ways (w1,w2) produced by the "split".
https://drive.google.com/file/d/1LNw96p6_stNMAPw8IXD3A08xDL_XHvX_/view?usp=sharing

r1 is the relation I deleted from w2
r2 is the relation someone deleted from w2
I don't know who eliminated r2, however I tried to analyze r2 as well, apparently it's wrong because apparently the right one is r3.
In conclusion I don't know who eliminated r2, if it was me, I did it unknowingly and without my being able to notice it, but I don't think it was a damage anyway because in my opinion r2 shouldn't exist because r3 already exists.
Did you fix anything after my "split"?
What is broken for you?

You are right, I will pay more attention in the future.

For this edit I used JOSM and I just "split" and removed a "way" from a hiking trail "relation".
Do you think I made a mistake? Or was there another problem?

I'm not sure, but it seems that someone has rightly removed the duplicate relations on a "way".
Perhaps the "split" corrected the situation automatically but only on one of the "ways" while the other "way" maintains the duplicate relations.
What do you think?

Excuse me but I don't understand, why do you say they are broken?
I did a "split" of a way which produced 2 "ways" which kept the relations:
way/223767595 (17 relations)
way/1116858964 (20 relations)
In fact, the difference is 3 relations (20-17) of which 1 I explicitly removed, the other 2 I don't know why he removed them, if you want I'll do a revert and try again.

Lo split di una way non elimina relazioni, non credo sia causa mia.

Come hai ipotizzato tu, probabilmente ho fatto uno split di una way per correggere la relazione "cammino delle dolomiti".
Qual'è il problema?

Ciao, come dice il commento sul changeset, ho corretto le ways come membri della relazione:relation/5944160
Non ho fatto altro su altre relazioni, nè ho creato o eliminato ways.

Con brouter web, selezionando "bici da trekking", crea correttamente il percorso, anche attraversando il ponte.

L'allineamento non è corretto in base alle ortofoto più recenti, vedi True Ortofoto RAFVG https://eaglefvg.regione.fvg.it

La way way/532565592#map=19/46.44955/13.25830 appartiene alla relation relation/140429#map=16/46.4534/13.2538
La relation ha il tag route=hiking mentre la way ha il tag climbing=route, questo mi fa sorgere il dubbio che ci sia un problema perché hinking e climbing non mi sembrano compatibili. In FVG questa è l'unica relazione con questo problema.

outdoor shop on the rocks?

Sbaglio, o questa modifica ha eliminato la parte di sentiero 358 che lo collega al 374?
https://hiking.waymarkedtrails.org/#route?id=140980&map=16.0/46.3303/12.4231

Please be careful not to remove "way" from "relation", see: relation/14068504/history

Regarding the Tabacco maps, from osm.org/copyright/en I extract this sentence: OSM contributors are reminded never to add data from any copyrighted sources (e.g. Google Maps or printed maps) without explicit permission from the copyright holders.

However, perhaps it refers to "S.R. UD 42 della Val Resia" which has nothing to do with this way. Try to see here: relation/1769146

What is ref=42 referring to?

Il ref=931 potrebbe essere un vecchio sentiero CAI?

Credo che questa modifica abbia interrotto 2 relations

Il sentiero 049a sembra che abbia una parte diversa da quella ufficiale CAI: https://www.cai-fvg.it/sentieri-cai-fvg/sentieri-cai-fvg-settore-0-carso/s0-049a/

sac_scale andrebbe nella way e non nella relazione