No, they don't have to say "The World" whenever they want to stop time.
You can find the reason why they said "The World" to stop time on this question:
What is the purpose of characters shouting out the name of their attack?
From the answer section, it's stated that,
It's a tradition where it was intended for young audiences to yell the
attack names with the character. The tradition began with Mazinger Z,
which is considered the first Super Robot anime. The producers of the
show felt that if the main character, Kouji Kabuto, shouted out the
attack names each time the mecha did them, then it would give the
target viewers, who were aged around 3 to 10 at the time, the ability
to literally join along in the fun.
The logic behind this is if the kids were to directly interact with a
show they already liked, then they'd like it even more and would stick
with the show in the long run. Needless to say, the strategy worked,
and literally all other mecha anime in the 70's (sans First Gundam in
late, late '79) copied the trend.
Thus, the tradition was born and still anime shows, regardless of
whether they are mecha themed or not, use it, regardless of their
From the user's answer, it's stated
Also, It seems that it was also done to make the shows feel more
intense. Something about the characters yelling their attacks just
makes the action better.
From what you said
when Dio attempts to stop time but fails since Jotaro manages to get
his heart working again quick enough.
The reason why JoJo could reach Dio's head even though Dio had verbally said "The World" to stop the time is that JoJo had a new ability to stop time at that time, but not that much, just about milliseconds. To prove that, these are some clips where JoJo tried to move when Dio stop the time.
Here are some examples to prove that Dio/Jotaro didn't have to say "The World" to stop time
JoJo's Bizarre Adventure: Stardust Crusaders
- When Polnareff attack Dio with Silver Chariot at 0:58
- When Polnareff meet Dio for the first time at 1:39
- When Hol Horse tried to kill Dio at 3:44
- When they tried to open Dio's coffin at 0:21
- Chase scene where Dio try to beat up Joseph 0:22
- Scene 1 0:23
- Scene 2 0:51
JoJo's Bizarre Adventure: Diamond Is Unbreakable
- Jotaro vs Jousuke 4:17
- Jotaro vs Rat 1 2:35
- Jotaro vs Rat 2 3:25