Definition of lead time and monthly averaged

Dear Forum,
I am currently extracting C3S monthly seasonal hindcasts spanning from 1993 to 2016 and I have a question regarding the terminology used for Leads.

Taking the example of the ‘burst’ model (ECMWF, CMCC, DWD, and ECCC), it is said that the convention used for labelling the data implies that leadtime_month=1 is the first complete calendar month after the initialization date. So, hindcasts initialised on the 1st of the month “M” with leadtime_month=1 is month “M”. As the forecasts may start after the first day of the first calendar month, the first calendar month (Lead 1) in principle, contains some assimilated observations and some period of pure forecast. This month thus represents in part the initial state and in part some forecast, but is not strictly either.
Is this correct?
(Also, when downloading the C3S data from the dataset: EU Copernicus CDS C3S leads are labelled from 0.5 to 5.5, akin to the NMME which employs a similar initialization method.)

Thank you very much,