"The president [Reagan] then was about 85 years old".
"And I swear to the Lord in the Lord’s house this is a true story. ... And the President -- true story -- President looked very sternly at Strom ..."
Ronald Reagan turned 85 in 1996 during the presidency of Bill Clinton. Biden in 2003 was off by nearly fifteen years.
The Thurmond-strong-arming-the-president anecdote, if it actually happened at all, must date to late 1982 because Reagan vetoed the Biden-Thurmond crime bill on January 15, 1983, which was what the meeting Biden recounts was about.
At about the 7 minute mark (the crowd is already skeptical of the previous anecdote's truthfulness at the 3:30 mark):