First of all that doesn't fit the man page:

It can test the right operand and become false on the same evaluation it became true (as in awk), but it still returns true once.

Secondly if you look at my third example above, even if the second operand is always false, nothing is printed. So it can't be a case of flip and flop on the same round, there never seems to be a flip

First of all that doesn't fit the man page

How does it not fit that quoted statement?

flip at 1? yes
flop at 1? yes      -> Would return false, but must return true once.
flip at 2? yes
flop at 2? yes      -> Would return false, but must return true once.
flip at 2? yes
flop at 2? yes      -> Would return false, but must return true once.
flip at 2? yes
flop at 2? yes      -> Would return false, but must return true once.
flip at 2? yes
flop at 2? yes      -> Would return false, but must return true once.
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Secondly if you look at my third example above, [...] So it can't be a case of flip and flop on the same round, there never seems to be a flip

It flips, but it never flops because $y is always false.

my $x=1;
my $y=0;
for(1,2,3,4,5) 
{
  next  if (printf("flip at %s? %s\n",$_,$x?'yes':'no'),$x)
        .. (printf("flop at %s? %s\n",$_,$y?'yes':'no'),$y);
  print "$_\n";
}
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flip at 1? yes
flop at 1? no      -> Returns true.
flop at 2? no      -> Returns true.
flop at 3? no      -> Returns true.
flop at 4? no      -> Returns true.
flop at 5? no      -> Returns true.
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Update: Truth tables for flip/flop:

..
State	Flip	Flop	New State	Returns
out	yes	yes	out	true
out	yes	no	in	true
out	no	[not executed]	out	false
in	[not executed]	yes	out	false
in	[not executed]	no	in	true

...
State	Flip	Flop	New State	Returns
out	yes	[not executed]	in	true
out	no	[not executed]	out	false
in	[not executed]	yes	out	false
in	[not executed]	no	in	true

Yes. you are right. I seem to have ignored the inverted sense done by the next.