The exit tax is a form of capital gains tax that applies to people who are moving out of Japan with assets worth 100 million yen or more who have been resident for 5 or more years.
The exit tax does not apply to people with employment-based statuses of residence. It does apply to people with spouse and permanent resident statuses of residence, and Japanese citizens.
(Work in Progress) Applicable residence statuses
|Exit tax applies||Exit tax does not apply|
|Japanese National, Permanent Resident, Spouse or Child of Permanent Resident, Long-Term Resident, etc.||Diplomat, Professor, Artist, Business Manager, Legal/ Accounting Services, Medical Services, Researcher, Instructor, Intra-Company Transferee, Temporary Visitor, Student, etc.|
Table compiled according to Q3 from 国外転出時課税制度（FAQ）（令和元年5月）(PDF) linked from 国外転出時課税制度 - National Tax Agency. This appears to be just a summary and the complete list needs to be taken from the immigration law.
The PDF mentions 所令170③一, which I found here but it doesn't seem to have an English translation. Quoted:
It is not clear what 上欄 corresponds to. There are 5 sections to the table it's referring to. If it's just the upper section, that's different to the examples in the NTA PDF. Maybe it just means "the stuff written in the table" rather than "the upper section of the table"? The "Upper part" (1) only contains these statuses: Diplomat, Official, Professor, Artist, Religious Activities, Journalist. It would be useful to know the rationale behind the groupings of table 1 in the immigration law, because all 5 sections are headed "Authorized Activities in Japan". The Immigration Agency site also uses this 上欄 terminology, but lists all the statuses. This list also has the category grouping explanations. The plot thickens. After reading this it seems that 上欄 can actually mean "the first column" - which makes perfect sense. Now just need to find a good source/examples to be satisfied that's the case, and this section can be edited properly.
Also: The rule says 在留していた期間を除く, or designated statues are not included as residence period for the purposes of the exit tax. This appears to imply that the 5 year timer starts from when you acquire a status that is subject not to the tax (as opposed to the tax being applicable immediately). It would be good to get confirmation of this point from an expert.