Octave wrong in low horn?

Using a MIDI keyboard, I entered the (treble clef) notes in the picture into horns of various pitches. I made sure that Edit > Transposed Pitch was selected before doing so.
Horn notes

For Horns in C (alto), B♭ (alto) and G (alto), Horns in F, E♭, D and D♭, the notated pitch stayed the same as I had played in, and the sound played corresponded to the pitch of the horn.

However, as I continued choosing horns of lower pitch (and playing exactly the same notes on the MIDI keyboard), the notation and the sound played jumped to an octave higher. Those which displayed this behaviour were Horn in C (sounds 8va bassa), Horn in B (sounds 8va bassa), Horn (basso) in B♭ and Horn (basso) in A.

Is this intended behaviour?
I found that to achieve the notation in the correct octave, I had to play the notes an octave lower ie in the actual sounding octave.

To the best of my knowledge and experience, when reading from the same notes, a Horn in B♭ basso should sound in the same octave as baritone horn (written in treble clef), tenor saxophone, euphonium and bass clarinet, namely a major 9th lower than written.

I think the playback of the displayed notation is in the correct octave. It just seems to be how Dorico interprets what is played in for the particular instrument - in this case, the low horns.

I followed up on this and entered a middle C in a C-Score in the C-alto Horn and copied it to the other horns via “copy to staff below”:

If I switch to transposed score, the pitches show correctly:

It seems like Dorico is handling the “basso” instruments similar to Piccolo Flute and Double Bass, where Octave-Transposition is not shown even in sounding scores. I don’t know if this is common practice, but seems unusual to me, too.

On another topic, it would be great if the “alto” or “basso” variant of the transposition could be reflected in the instrument name, such that “Bb (basso)” is the full description of the transposition.

As it stood now, I had tow Horns in C, one alto and one basso, which were also numbered 1 & 2, but which are totally different instruments, just as different as a Horn in C and a Horn in D, which are not numbered. Including the full transposition would fix this issue, and is necessary anyhow, IMHO.

I had entered the notes into each horn separately by playing them on a MIDI keyboard. I did not do any copying or pasting between instruments. It may well be the case that once the notes are on the page they are correct and will copy/paste correctly, but my concern was based on the notes that I played into the horn in B♭ basso (and the others mentioned) sounded and were notated an octave too high. Baritone horn and tenor saxophone sounded and were notated in the correct octave.

I agree on your point about the transposition being included in the instrument name.

My point was, that they don’t copy-paste correctly, as the transposing score shows 3 middle Cs and 2 an octave higher in the sounding score. All notes sound the same in my example, too.

I’m loath to get involved in this at the moment, but the short answer to the question:

…is “yes”. We have set up those horns such that they are written an octave higher than they sound, like a double bass. I can’t remember (and may never have known) the precise source we would have used for this information, but we have at least one professional hornist among our beta testers, and I feel reasonably sure he provided some feedback on this complete set of horns.

You can always provide your own (clef and) transposition overrides using the provided feature for that purpose.