Please give a complete overview, how the devices are connected in the two described use cases.

Generally spoken, it makes no sense to connect different headphone amplifiers in series.
Usually, sound becomes worse, the more components are involved in the signal path.

And you can see already here, that you do not get it handled to keep the signal on a proper level.

For more clarity of what you are doing, please provide such a drawing to make it easier to understand.

Also note, whether a connection is analog or digital and what reference level is configured on analog inputs and outputs.

This statement makes absolutely no sense to me:

I am confused because the Lehmann has no SPDIF output.
So if you talk about connecting the Lehmann to the ADI-2 DAC via RCA ... the ADI-2 DAC FS has no analog inputs with RCA plugs. This is a SPDIF (coax) digital input.

Awaiting a more clear description of the two setups that you talk about.

Would be interesting to know how the tape-out of this amp is circuited, a block-diagramm would help...
What model integrated amp is this ?

Oh what a great amp !
If you pull fully down the volume on the E380, is the headphone silent or still have the same volume via tape-out ?

To be honest, I still don't understand your setup, a drawing would provide a better overview.

I also have an Accuphase amplifier (E-600) and I have a clear concept.
A recording interface as the basis (UFX III) and then an ADI-2 Pro FS R BE for recording and the hi-fi corner.
Headphones are connected to one of the two ADI-2 Pro FS, in my case, always in the recording corner.

Audio transport is digital from source to destination and, above all, lossless, validated by bit test (ADI-2 DAC/Pro feature).
The D/A conversion always takes place on the ADI-2 Pro, unadulterated, and thus delivers optimum results.

Various reference levels and the Auto Ref level and dynamic loudness features also provide further good features to keep SNR as high as possible and prevent the “sound” from thinning out even when listening quietly. It's a well-rounded concept.

In my opinion, all this fiddling around trying to interpose something as a “sound enhancer” is unnecessary and runs the risk of degrading the sound.

PC (audio player, MusicBee with ASIO support)
| \
|  +--USB2---------------------------------------------------------------------------------+ ADI-2 remote SW
|                                                                                                                     | (remote control of ADI-2 Pro via USB)
| USB3                                                                                                            |
|                                                                                                                     |
UFX III---ADAT1 (Main Out)---------[SPDIF]-------------------------->ADAT IN---ADI-2 Pro FS R BE----1/2--->Geithain RL906
\   \                                                                                                                                  \------------3/4--->Audeze LCD-3 / AEON 2 CB
  \   \                                                                                                                                        (Monitoring Studio Corner)
   \   +-----ADAT2 IN<-----------------[SPDIF]-------------------ADAT OUT--+ (*)
    \                                                                                                       \
     +-------ADAT2 (Main Out B)------[SPDIF]------>Oehlbach------>ADAT IN---ADI-2 Pro FS R BE----1/2--->Accu E-600
                                                       BluRay------>Optosel 4:1                                          \                         /
                                                    OLED TV------>Mk II TOSLINK                                         +<----AN IN----+       
                                                     Reserve------>Switcher (**)                                (HiFi Corner or Speaker B for UFX III)

(*) Just in case I want to record sound from TV coming from E-600
(**) The TOSLINK switch supports up to 192 kHz. It even works with 15+1m TOSLINK cable. Can be controlled via Remote.
        By this TOSLINK switcher I have something like an ADI-2 Pro with 4x ADAT inputs.
        This way, I use the ADI-2 Pro as D/A converter for all digital devices in the HiFi corner.
        Additional benefit by using TOSLINK connections -> galvanic isolation (otherwise you get hum from Sat/TV).

The tape-out of the E380 seems to be a straight forward of the input signal.
No improvement of the signal, the only reason to connect the E380 into the chain is to be able to switch the different inputs with the big knob on the front. Select CD or Aux or Phono etc. to play out on the tape-rec.
I would connect the headphone amp directly to the audio interface only and make all level settings to fit.

+1

I still do not understand it.

Example:

> Recently I did receive a headphones amplifier, Lehmann Linear II, actually this amplifier is plugged
> in to the integrated, Tape Rec,

What is "the integrated" / "Tape Rec" ... Where ? What ? At what position in the chain.
Maybe its easy for you as you build it up so you are and live in "your wording".

But as this is no typical studio setup.
We do not know your devices, their capabilties, we do not know how this all is physically connected to be able to follow you what the signal flow of "A" and "B" is, what you compare.

It would be kind (not to say necessary) to make a drawing so that the setup is understandable.
You know, "a picture tells more then thousand words" ;-) Thanks!

Here’s the point:

You can’t just connect/combine two outputs, E380 RCA out AND ADI-2 RCA out into one input (Lehmann) through a splitter.
One output shorts the other, doesn’t work properly as you found out.

Either replug the RCA-cable as a one-to-one connection for each purpose,
or get a 2-way passive (no electronics) switchbox to add input switching capabilities for the Lehmann:

EXAMPLE PICTURES: