Naval custom of thirteen months.

The link that johncole provides does mention the “nautical day” as starting at noon but also says this was ended in 1805 and before 1805 it seems it was not clear or universal, rtaher it was used for log keeping only and was a source of confusion which is why it was ended.

So, yes, it seems from that cite that for an indetermined number of years before 1805 some or all British ships used a nautical date which started 12 hrs before the local civil date and they used both simultaneously which was a source of confusion. I do not think the custom was widespread in other countries or in merchant ships. What was universally widespread in merchant and naval vessels, British and foreign, was the practice of reckoning the days run from local apparent noon to local apparent noon and of adjusting the begining of the afternoon watch.

Different customs were kept with regards to changing the date (date line) etc. It used to be that the Philippines was on the same side of the date line as south America because it was colonized by the Spanish from there so an English ship out of Hong Kong and a Spanish ship out of Manila would meet and reckon dates a day apart.