Beautiful ships! I take it, the first project is also from early 1930s? What kind of guns they were supposed to be armed with?
It is a development of an OTO battlecruiser project, dated to February 1933. It is probably a development of a smaller - 27,200 tons standard - battlecruiser design from from November 1932. As an aside - it is interesting to note that OTO is the only entity that actually developed any designs explicitly as 'battlecruisers' in Italy in the interwar period, with all other designs developed by the RM or MARICOMINAV being battleships, even if small ones.
The main gun is a 343mm/53 - though it is hard to say more about the gun than that. The 343mm caliber was popular with many of the Italian 'intermediate' battleship concepts of the early 1930s, when they were exploring smaller capital ships as contingences for any of the international treaty negotiations that might have barred the favored 35,000-ton type (such as the British proposal to limit battleships to 25,000 tons, or the negotiations to the French for a potential cap of 26,500 tons).
In particular, it was a feature of several of the battleship concepts that were put forward as a potential purpose-built response to the French construction of
Dunkerque, until the definitive decision to rebuild the
Conte di Cavour-class battleships (in November 1932). Though the gun caliber chosen for the RM's in-house battleship design (29,500 tons standard) is not known, both private industry offerings - the Design 770 battleship and the 26,500 ton battleship (not battlecruiser), also shown in the OP's post - were armed with a 343mm gun. Design 770 came from Ansaldo, unfortunately it is not clear where the 26,500-ton design came from - but it would have to be either OTO or CRDA.
Given OTO was developing its own battlecruiser project (the aforementioned Type F) in this period, I would tend to assume the 26,500-ton battleship would have been CRDA's effort? It's not impossible that it is an OTO design either, however, and it seems OTO was developing multiple designs in this period - there is for example another project of theirs, also explicitly called a battlecruiser (26,670 tons, 3xIII 381mm guns), that survives as a model at the Naval Technical Museum of La Spezia. It is another design that, according to the placard, was developed between 1930 and 1935 in relation to ongoing efforts to rebuild the Cavour-class or the construction of new battleships.
Regardless, neither the Type F / Design 134/1 or it's 27,200-ton predecessor were offered as an alternative in the November 1932 discussions for the
Dunkerque counter, and nor was the 26,670 ton design.