What the age of a person required to be mentioned for all purposes is in completed years from his /her officially recorded date of birth only. Logically also that is correct for a child can be called one year old only on its first birth anniversary. Therefore, present age of a person is the no. of years lapsed from his date of birth.
The person cited in your example would be completing his 18th year of age on 01-06-2018. So, as on 2nd June 2018, he would be 18 years old though actually he is running his 19th year of birth.
From India, Salem