In the US, this variation is more popular than the original. Terrence has been in the US Top 1000 since 2013, Terence has been left behind.
In England and Wales, the opposite is true: Terence is at #282, whereas Terrence is not within the Top 2000.
Terrence Malick, American film director
Terrence McNally, American playwright
Terrence O'Hara, American TV actor/director
Terrence Dashon Howard, American actor
Terrence Hardiman, English actor
Terrence Mann, American stage actor