Yes, Judit, the balanced connection has a +6db difference, if I remember correctly.
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Actually, the difference in output between single-ended and balanced connection will vary depending the gear in question, but the increase with balanced is more typically on the order of about 3dB or so (6db means twice as loud), which sounds more in line with Judit's experience of having to turn down two 'clicks'. It is true that the common-mode noise rejection will improve by up to about 6dB.
From published measurements I have seen over the years, there is no 'exactly' or 'always' about these things. But we don't as yet know whether Judit's CDP is a true balanced design throughout, or just uses a splitter circuit to give a 'balanced' output. If it is truly balanced, than that will be the reference output level, and a single-ended output will be lower in level, maybe by closer to the 6dB figure than if it is not truly balanced (most CDP's are not). The common-mode noise rejection has mostly to do with the RFI/EMI picked up on the connecting cable. But I'm like you on these matters - definitely open to correction! The main thing is if Judit prefers the sound with the new connection.
The Electrocompaniet CDP is a true balanced design, as is the Pass Labs X-1 Pre that is next in the chain.
I have been swapping in new PCs as well. Since I do not have dedicated line I have a ground loop problem which I currently correct by floating all but one of my components. I have noted that polarity (which way I plug the floated PCs into their outlets) affects output SNR as well.
Oddly enough, even Discovery Essence RCA interconnects seem to have a lower SNR (than other RCA interconnects) because I also tend to turn DOWN the volume knob a couple of notches when I use Essence RCA IC's between my CDP and pre-amp.
This is compared to other IC's such as Shunyata, Aries; Coincident, CTS; and HT, Truth Link. Go figure.