[by'dʒeː] -> Online-Duden, Pons, Collins, Muret-Sanders

The transcription of the German word in OP is a non-phonemic transcription. The symbol [ʥ] represents a more palatalized consonant, but I think no one would claim that /ʥ/ or the second element of it was a phoneme of German. In a phonemic transcription, there would be no need to distinguish it from [dʒ]. The purpose is to assert that the German sound is more palatalized than the English one (whether this is true, I'm not sure). However, as a rule dictionaries give phonemic transcriptions (and should then indicate separately how the phonemes are pronounced).

Take a simple example. There is only one /l/ phoneme in English, but in RP (at least) there are two major varieties (velarized ('dark') and non-velarized ('clear')) plus other distinctions (voiced and voiceless). However, because all these differences are predictable from context, a phonemic transcription would render them all the same: /l/. A non-phonemic (i.e. phonetic) transcription would distinguish the varieties in more or less detail as the occasion demanded.

In the present case, there's no certainly phonemic distinction between the two sounds in German, and I doubt they both occur anyway. This being the case, the normal practice would be to use the symbol which is most widely used internationally, and indicate (if required, and if true) that the German sound is more palatalized than the English one. (I notice that the Wordreference transcription is given in square brackets, which by convention indicates a phonetic, not a phonemic, transcription.)

Ich bin kein Linguist und kann das vermutlich nicht korrekt beschreiben. Für mich klingt das englische "dsch" ein bißchen härter als das deutsche:

...

was vielleicht mit den beiden Vokalen zusammenhängen könnte: im engl. ein a und kurzes e, im Dt., dem Frz. folgend, ein ü und ein langes e

@#7

'härter' means different things to different people, but I think I know what you mean. There is less audible friction in the German sound. I think this is what the Wordreference transcription is trying to convey. While the different symbols officially relate to the place of articulation (that's the way the IPA works), the acoustic effect is as you describe.

There is always a problem of transcription when the sound occurs in borrowed words only, not least (but not only) because the pronunciation is likely to be unstable.