Japanese people don't care much about it. For example 沖田総司 Souji Okita a member of 新選組 Shinsen-gumi died in 慶応4年5月30日 = July 19th 1868, this is probably your question.
Japanese calendar is confusing. Before the internet, most people forced the imperial calendar. Now many people found inconvenience and shifting into western calendar.
Imperial calendar(元号): The new period starts with the new emperor. 平成 1989 - 2018, the current emperor told us his abdication on 2019. It is going to be a new period.
Any documents from Public office are written in this way.e.g Driving License expire 平成31 and I have no crew because I don't remember year in 元号. My grandmother was born in 大正15年 hard to count her age because of sum of 平成, 昭和 and 大正, it is very simple to calculate if she could tell us born in 1926. Most commercial companies no longer use Imperial calendar, Insurance companies, Bank and Stock companies. They were fed up with the problems became real after the shift of 平成 new imperial year such as no 昭和105年 and etc. News sometimes tells us "The new bullet train, the linear motor car" plans to start commercial drive in year of 30!" (sorry it's strange English but no other way to tell) So people heard this it comes in a years or 13 years. This is annoying.
Lunar calendar 旧暦: it's different from Chinese calendar and roughly a month or two late from western calendar, as you know it's based on moon face, the full moon day is 15日, the new moon is 1日, month ends 29 or 30. This had been s an official until 明治5年12月2日 = December 31th 1872. The installation of Gregorian calendar in Japan was 1873 January 1st. So Meiji 5 has 26 days shorter than other years. Before this happens every documents in Japan were written in Imperial year and Japanese lunar calendar.
Japanese lunar calendar is somewhat fuzzy. It's very different from Chinese.
In some suburban places, they has been hold ひな祭り(hina-matsuri) March 3th, it was on the lunar calendar, so they found the way it's roughly a month late then it'd be held on April 3th.
Before the Meiji period, it would be the lunar calendar with imperial year which is very hard to know the accurate date in western calendar.
For year 2017 = 平成/Heisei 29 = 昭和/Showa 92 = 大正/Taisho 106 = 明治/Meiji 150
For Month in modren/old Japanese alias