The voice is the only instrument where you can "play" (sing) what note you think. So, the advantage of absolute pitch for improvisation is minimised.
However, AP would be advantage in vocal sight reading. With AP, you will never sing a wrong note, whereas a non-AP could make some mistakes, depending on how strong there musical ability/relative pitch is.
I am not sure shifting keys in singing would be hard for people with AP. Of course they would be aware of the exact new notes they would be singing whereas nonAP would simply thing "everything is X steps up/down" but relatively the same.
I don't think AP has an effect on tolerance to listening to a song in a different key, it is more your personal taste. I don't have AP, but I can tell when a song is in a different key from the original. I find it acceptable, so long as it is in tune. But I vastly prefer the original key simply due to familiarity. Also, some songs really do sound better in certain keys than others.
How so? I can definitely sing random pitches that don't conform to, say, 12-tone equal temperament. If I sing into a guitar tuner, I can make the meter move continuously from its flattest to sharpest position.
Their point is that there are dozens of instruments that also fit that criterion; i.e. the human voice is not the only instrument with that capacity.
Fretless sting instruments allow for continuous pitch modulation - that includes the violin family as well as others like pedal steel guitars, the Japanese shamisen, etc.
Certain wind instruments like slide trombones and slide whistles do the same.
There are also electronic instruments like the theremin or any synthesizer with a pitch bend wheel.
Audiation is a basic skill for any instrument. Accomplished musicians are trivially able to improvise complex melodies that they can play and sing simultaneously.
I'm a jazz guitarist who has developed the very bad habit of singing what I play while I improvise. Working hard to break it. But what I sing, and what I play is the same.
Being able to "think it, play it" is absolutely central to what I do when I improvise.
However, AP would be advantage in vocal sight reading. With AP, you will never sing a wrong note, whereas a non-AP could make some mistakes, depending on how strong there musical ability/relative pitch is.
I am not sure shifting keys in singing would be hard for people with AP. Of course they would be aware of the exact new notes they would be singing whereas nonAP would simply thing "everything is X steps up/down" but relatively the same.
I don't think AP has an effect on tolerance to listening to a song in a different key, it is more your personal taste. I don't have AP, but I can tell when a song is in a different key from the original. I find it acceptable, so long as it is in tune. But I vastly prefer the original key simply due to familiarity. Also, some songs really do sound better in certain keys than others.