The phrase "ex-con" or "ex-convict" is well established to refer to a person who has been convicted of a crime and who has completed any sentence associated the crime of which they were convicted, distinguishing them from both those who are still serving a sentence and from those who have never been convicted of a crime.

(Its derived from the usage of the term "convict" to refer specifically to a person who is currently serving a sentence for a conviction of a crime, i.e., a prisoner or inmate.)

If "convict", the noun, had a primary definition of anyone who has ever been convicted rather than someone serving a sentence for a conviction, then, yes, you might be right that you couldn't be an "ex-convict". But that's not what the word means.

"Ex-con" refers to "ex-convict", a former inmate in jail, not "ex-conviction", one who has been previously found guilty of a crime.