Current adder or voltage adder

Both adders (summers) were invented by Mr. Kirchhoff as early as the 19th century. Both are genius simple because they are made of... nothing. The current adder is just a node and obeys the KCL; the voltage adder is just a loop and obeys the KVL.

Diodes are nonlinear elements and have nothing to do with these linear analog "devices".

You can visit two Wikibooks stories about the voltage summer and current summer to get familiar with the philosophy of the problem.

Firstly your picture

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Voltage adder

Within constraints it's a pretty decent $boxed{text{voltage multiplier}}$ but, not a voltage adder unfortunately. So, using two diodes you can get a pretty good linear signal multiplier (modulation): -

The raw output looks like this: -

The red signal trace is the modulation; a 10 kHz triangle wave and, the blue trace is a sine-wave carrier at 1 MHz. The diodes allow this sort of thing to happen.

And, if you band-pass filtered that signal you get good old-fashioned DSB broadcast AM: -

Circuit with 1 MHz band-pass filter as per the above: -

Other than that it will also add currents because there is nowhere else for the currents to go except to add at the common node.

Neither, if the inputs come from voltage sources. The voltage at resistor output would equal to whichever of the two input voltages is higher, minus the voltage drop of the diode. Of course, if the inputs are not fed with a voltage source, the situation is more complex. If the inputs are driven with current sources, then the currents will add up, but currents would add up even without the diodes so they are irrelevant.