Regarding the problem- issue of "pushing a string into a straight line" consider the following thought-experiment.
Imagine a piece of string lying on a tabletop. Then imagine putting a pin into one end of the string to keep its position static. One can then straighten that string by pulling on the other end until the string is straight. Alternatively, one can place one's fingernail a few miilimetres from the other end of the string and push the string straight by pushing that string end against the tabletop in a straight line direction away from the pin.
C'mon, Jeff,
Both motions in your examples are called pulling the string!
Pull Definition:
To exert force or influence on so as to cause to move toward or after the source of the force; drag, tug, draw, attract, etc.
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