That's not exactly how it works. The House tried and convicted Clinton, which is why he is called the impeached president. The Senate assesses penalties and they refused to assess any penalty on Clinton which was why he was never removed from office or assessed any other penalty as the result of being impeached.
The last part of your post is dead on!
“That’s not exactly how it works. The House tried and convicted Clinton, which is why he is called the impeached president. The Senate assesses penalties and they refused to assess any penalty on Clinton”
This is not true. An impeachment is an indictment, an accusal, not a conviction. The same body would not indict and convict. That would be a violation of common law. It is up to the senate whether the accused is guilty or not guilty of the accusal. In the case of Clinton, they found him not guilty. Should 2/3rds have found him guilty, he would have been removed from office immediately.
Article I, Section 3, 6:
“The Senate shall have the sole Power to try all Impeachments. When sitting for that Purpose, they shall be on Oath or Affirmation. When the President of the United States is tried, the Chief Justice shall preside: And no Person shall be convicted without the Concurrence of two thirds of the Members present.”
The Senate tried Clinton for perjury and obstruction of justice, but they did not get the two thirds vote to convict him.