Here is a site that offers some resolution to the origin of the abacus.
This is Ryerson University, in Toronto, Ontario, Canada.
This site makes a distinction between the ancient counting boards, and the "abacus" the Chinese invented.
"The difference between a counting board and an abacus."
It is important to distinguish the early abacuses (or abaci) known as counting boards from the modern abaci. The counting board is a piece of wood, stone or metal with carved grooves or painted lines between which beads, pebbles or metal discs were moved. The abacus is a device, usually of wood (plastic, in recent times), having a frame that holds rods with freely-sliding beads mounted on them.
Both the abacus and the counting board are mechanical aids used for counting; they are not calculators in the sense we use the word today. The person operating the abacus performs calculations in their head and uses the abacus as a physical aid to keep track of the sums, the carrys, etc."
The oldest surviving counting board is the Salamis tablet (originally thought to be a gaming board), used by the Babylonians circa 300 B.C., discovered in 1846 on the island of Salamis. There are references to them going back much further of course.
The abacus as we know it today, appeared (was chronicled) circa 1200 A.D. in China.
One can argue that the two are related, or not. Often, history has shown that inventions arise independently in mulitple times and places. Historians, seems to want to only give credit to one (usually their own).