In Belknap’s case, at least, the reason given by most of those who voted to acquit was that the case had become moot, not that they lacked authority to ban him from future office.
“Conviction” only in the context of the impeachment process. The Constitution also explicitly separates it from the realm of actual criminal law. Also, remember that impeachment can be for anything the House wants to impeach for, not necessarily actual crimes. One of the charges against Andrew Johnson was for making disparaging speeches about Congress, for instance.
Alcee Hastings, since he’s been mentioned already, was removed as a judge for corruption but was not barred from further office, in recognition of the inconvenient fact that he had actually been acquitted at his criminal trial. He did take the money, though, that wasn’t much in doubt.