i believed he can perform vs i believe he can perform

Some examples from the web:

• With that done, I was free to break out the product I'd be promoting, the Corby trouser press, which I believed could perform a vital extra service for the Indian customer.
• He believed the services you perform.
• The Men of Letters believed that you could perform a tracking spell with extracted Grace, but they were never able to test the theory.
• I believed he saw promise in me, but apparently not.
• And if I believed he had a plan...
• Gone all the time, and I believed he was working.
• I shot him because I believed he was about to shoot me.
• I don't know. I believed he cared for me.
• But at that time, I believed he sent that boy to teach me a lesson.
• Did he go into detail about s might be or where their bodies might 0FFFF} No, he didn't, but I believed he was ose to it.
• And for a long time, I believed he would change... finally get sober, finally marry me.
• Lorta believed he was privileged to live here.
• Your mother and I believed Krypton lost something precious.
• I believed they returned to punish our family.
• I believed it on impartial conviction.
• I believed that the Sisterhood could survive anything.
• I believed work was all that mattered.
• I believed every word he said.
• He believed he was chosen by destiny to rescue Germany.
• Hitler believed he was chosen by destiny to destroy the Jews.

Some examples from the web:

• Some deploy superior research and stock picking skills to buy the best ideas and then short those they believe will perform less well.
• 1.14 The Commission does not believe that NGOs perform the role of a public authority.
• I believe he'll arrive momentarily.
• He doesn't need you to perform make-believe medicine, right, Michael?
• I believe he's your new boss.
• I believe he truly regrets his lot.
• I believe he was known in the neighborhood as Blue.
• I believe he has met him, but his description is vague.
• I believe he worked in your unit.
• I believe he intended to capture and drown the princes.
• I believe he's under the impression that his sister was successful.
• I believe he's in the transporter room, Captain.
• I believe he's in diabetic ketoacidosis.
• I believe he's speaking English, Sir.
• I believe he means the device that camouflages our ship has ceased to function...
• I believe he left, sir.
• I believe he stopped here minutes before he was killed.
• I believe he's a card cheat.
• I believe he walks among us... taking many forms.
• I believe he needs very powerful protection, Tavi.