You are absolutely correct. It is an Alto Clarinet, more usually referred to as the Basset horn.
Clarinets are complicated. The development of the clarinet is inexorably linked to Mozart and his friend Anton Stadler.
The clarinet that Mozart actually wrote for was the Basset clarinet. The modern soprano clarinet goes down to E and E flat if you are lucky. The Basset Clarinet goes down to C. It is now becoming much more usual to perform Mozart's works with the Basset clarinet and they sound much better that way.
Towards the end of his life he became more interested in the Basset Horn and wrote for it.
The development of the oboe is even more complicated, but we won't get into that, except that in times gone by there were are far greater variety of oboes than there are today, with many of them like the Oboe da caccia having no real modern equivalent in terms of sound. Bach in particular wrote for every type of oboe around, especially the oboe da caccia and the oboe d'amore.
Oboe da cacchia.
Oboe d'amore
I'm happy to report that these instruments are again being used in performance. No serious recordings of Bach's orchestral and choral music does not employ these instruments and others of the period.
