If you withdraw a LinkedIn invitation, the 3 weeks you have read about is LinkedIn's own figure. Its help page on withdrawing invitations says you cannot send a new invitation to the same member for up to three weeks. "Up to" means LinkedIn gives no end date. The block covers that one person, and everyone else you might invite is untouched.
What withdrawing does
The person is not told that you withdrew it.
LinkedIn's reminder emails to them stop.
It cannot be restored, so a slip of the thumb costs the same wait as a decision.
The decoder above says to try again later for this case: another invitation to that person is possible, just not yet. Pressing Connect before the wait ends gets refused again. The decoder offers no way around the wait, and neither does this page.
When to withdraw, and when not to
Withdraw an invitation that has sat pending for weeks when you no longer expect an answer. LinkedIn names invitations left pending as a reason it stops accounts sending more, so clearing old ones helps. Withdraw one sent to the wrong person, too. Either way, it is one at a time.
Do not withdraw to fix a note or to make an invitation show as new. The corrected one is refused until the wait ends, and the person probably never read the first one closely anyway.
What to do in the meantime
Write down the day you withdrew. LinkedIn gives no end date, so your note is the only record of when to look again.
Reach the person another way: follow them, or send an InMail if your plan has credits.
After the wait, invite once, with a reason to connect. If LinkedIn refuses again, stop there.
The LinkedIn invitation limit covers which pending invitations to clear and why they count.
In a Oneprofile sequence, a request LinkedIn refuses after a recent withdrawal fails with that reason and the date to retry. A failed step never runs again on its own.