BBJ1, 73W, BBJ2, 73H etc. are either IATA or commercial names.
Log Book entries should be ICAO recognised designations, e.g. B737-700, B737-800 etc. The same goes for airport abbreviations, use ICAO, not IATA, for example, at Amsterdam don't enter AMS, enter EHAM.
In the end it's up to you, but the regulatory authorities who might inspect your log book pray to the ICAO God.
Old Smokey