Yes I understand what you mean, and to clarify the exercise uses the classical melodic form (MM ascending, Aeolian descending)
If the 6th and 7th are Major intervals in both the ascending and descending form, I would call it the Jazz Minor scale to avoid confusion. But of course not everyone knows what Jazz Minor means, so I would probably have to explain

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It's not a matter of language, it's matter of context and semantics. English speakers also quibble over whether the 7th or the tonic is the last note of the scale.
This is my preferred usage in this specific context. The scale is played tonic to tonic (G to G). So G is the last note of the ascending G Melodic Minor scale, as played. You would not normally stop on the 7th, you would play through to the tonic.
However, if you asked me a theoretical question, "What is the last note of the ascending G Melodic Minor form?" I would say F#. When you are talking about the notes, you don't have a need to restate the tonic.
Hopefully you understand what I mean, even if you don't agree.
Hopefully this will make everything clear: After you play the Melodic ascending form, you go up a half staff and play the descending (classical) melodic minor form. In other words you play the Aeolian or natural minor form.
Is G not the leading tone in Ab Melodic Minor? Here is what is written in the exercise: G->Ab Gb Fb Eb Db Cb Bb Ab, etc. But remember we are using the classical form of the Melodic scale.
Ab then becomes the (enharmonic) leading tone for the next ascending melodic scale: Ab->A B C D E F# G# A
Your English is fabulous by the way, although I am not surprised since you are in Germany

. Tschüss.