The word "balistrlis = crossbow man" reminded me of the Balearic Isles, probably so named
because the Romans got most if not all of there slingers from there.
Here is a site showing sling and way it was used. I have used a sling, and I think the picture
of throwing underhanded is in error. I swung the sling horizontally, with a sudden acceleration, a
snap of the forearm just prior to letting go of the cord held between thumb and index finger.
The other cord had a loop aroung the middle finger.The effect of the sling is to greatly increase
the length of the arm. Even when an early teenager, I could throw a metal roller bearing from a
wrecked automobile well over a hundred yards. I had a site showing pictures of the missiles the
Calearic slingers used, of molded lead about two inches long, a half inch in diameter, pointed at
both ends, weighing up to two ounces. They have been found in large numbers at old battle sites.
Montesquieu says one of Hannibal's victories over the Romans followed the Roman general's
being hit by a missile from a sling.

From Montesquieu:
Hannibal's light troops were in a different class from those of the Romans. They
were very highly trained and were capable of inflicting much damage on the enemy.
Of particular importance were the Balearic slingers. These were organized into two
corps, each a thousand strong. They were armed with three types of sling for
employment at different ranges. Such was their accuracy and their volume of fire
that they were deemed more useful than archers (pages 23-24).

It is very likely that Hannibal had posted many of his Balearic slingers directly
opposite the Roman cavalry quite deliberately to disrupt both men and horses
(page 80).

The term catapult comes from the Greek word "Katapeltes" "shield piecer" (kata = through,
pelta = small shield"). For a picture of a reconstruction of one, see:
http://www.legionxxiv.org/lrgballistacrew/

The sling you describe would be the one David used to dispatch Goliath. We often seem to imagine it as some miracle that someone with what we think of as a child's toy could best the champion of the opposing forces. The sling was the single most effective personal weapon in accuracy and deadliness until the invention of probably the crossbow.

But the English long bow was deadlier than the crossbow, because it took so long to crank
up the crossbow. Remember Agincourt and Crécy. I remember reading that at one of those
battles, the French crossbow men, who had to use both ands to draw the bowstring back
with the bow held against the ground by their feet, were seriously hampered by ground
being muddy. The longbow could shoot perhaps six arrows in the time the crossbow could fire one.
Incidentally, I wonder what the "yew", which was the preferred wood, looked like. I had
evergreen yews almost twenty feet tall in my yard, but none of them would have had straight
grain necessary for a six foot bow, unless laminated, which I doubt that longbows were.

That's why I said probably. I wasn't sure about the relative dates of the crossbow and the long bow. Certainly the rate of fire on the sling was greater than that of the crossbow. The crossbow had more penetrating power and was therefore probably deadlier than the sling.

I searched for a found ads for yew longbows. Cheap at $600 ! And wood was said to be
Taxus brevifolia, which must be similar to the yews I used to own. But mine were all
too curved to be suitable. My father made us bows of hickory, which seemed to work
quite well.