What should be the proper method to build a LINE-OUT signal from a Synthesizer that has only "Speaker" and "Headphones-out" outputs?
Figure 1. Voltage vs. time of sine waves at reference and line levels, with VRMS, VPK, and VPP marked for the +4 dBu line level. Source: Wikipedia Line level.

The most common nominal level for professional equipment is +4 dBu (by convention, decibel values are written with an explicit sign symbol). For consumer equipment it is −10 dBV, which is used to reduce manufacturing costs.
Expressed in absolute terms, a signal at −10 dBV is equivalent to a sine wave signal with a peak amplitude (VPK) of approximately 0.447 volts, or any general signal at 0.316 volts root mean square (VRMS). A signal at +4 dBu is equivalent to a sine wave signal with a peak amplitude of approximately 1.736 volts, or any general signal at approximately 1.228 VRMS.

Based on the article above I would do the following:

Set the volume at about 75% volume (so that you have a some headroom, if required).
Select an organ or flute tone that will give a continuous tone without too many harmonics.
Play an A1 which should be 55 Hz and is close enough to mains frequency to give a reasonably accurate reading on a true RMS multimeter AC range. Measure the output voltage between one channel and ground.




simulate this circuit &ndash; Schematic created using CircuitLab
Figure 2. The attenuator &quot;pad&quot;.

Calculate the attenuation required to get the output down to 0.316 VRMS.
Use the attenuation value to calculate the values of R1 and R2.

You're generally going to be feeding high impedance equipment so R3 being 1k rather than 600 Ω will make little difference other than making the maths easier.
Note that it's not a true line out as the signal level will be dependent on volume control setting.