The "quarter", "half", and "full bridge" terminology stems from the strain gauge world, and specifies how many of the four bridge elements vary as the load cell is strained.
When Wheatstone came up with the "bridge" idea, he was trying to compare an unknown impedance to a known one, so only the balance aspect was important.
When you build a weighing scale using a "quarter" bridge, then you have the complexity of linearizing the output curve (see the simulation results, post #2 and #5), as well as having to regulate the excitation voltage to prevent that from effecting the slope of the output function.
A better way of building a weighing scale is the use a "half bridge load cell" (where two elements vary, usually one increases resistance while the other decreases), making the function more linear, and partially cancelling thermal effects.
An even better way of building a weighing scale is to use a "full bridge load cell" (where all four elements vary, two increasing while the other two decrease), making the function still more linear, and better cancelling thermal effects.
When interfacing something like an LDR or thermistor to the AD input of a uProcessor, folks refer to that as a "half bridge", but IMHO that is not a bridge at all; just a resistive voltage divider made out of a fixed resistor and a variable resistor. (Has nothing to do with Wheatstone or bridge).