I suggest using this page as an ear training exercise. In the case of the "baggage" underneath Jins Rast, listen for the tonic, and then listen in the melody for the scale degrees underneath the tonic. In the case of the Jins Baggage above Jins Ajam and Jins Rast, listen for the 5th scale degree above the tonic (the traditionally defined upper boundary of these ajnas), and then listen for the notes above that within the melody--and verify for yourself that you still identify the same tonic, despite the melody having gone up to the 6th scale degree above it.
The "quarter-tone" note a step beneath the Rast tonic is a defining feature of this Jins. Whenever Jins Rast appears, this note/interval usually comes along. Melodies within Jins Rast occasionally dip a step below that as well, to scale degree 6 below the tonic.
From Samai Bayati il-3ariyan, #16
From Nura Nura, #13 & #20
From Ba3eed 3annak, #30
It is very common for both Jins Rast and Jins Ajam to be signalled by the prominent use of the raised (or natural) 6th scale degree above the jins tonic, when preceded by another jins such as Hijaz or Bayati on the 5th scale degree above that tonic, which includes a lowered (by flat or half-flat) note in the same position (2 relative to the Bayati or Hijaz, 6 relative to the Rast or Ajam a 5th lower). In other words, the natural 6 results from raising the lowered 6, and in a sense "erases" the feeling of Hijaz or Bayati, and leads us melodically down to the Rast or Ajam tonic. This phenomenon occurs both when Rast and Ajam are the primary jins of a Maqam, as well as when they are themselves secondary ajnas.
From "Taheya" dance instrumental, #11 & #13
From Muwashshah "Sihtu Wajdan", #18
From Hayrana Leh, #14, #19, & #31
From Aruh Li Meen, #13
From Lissa Fakir, #04, #09 & #24
From Ya Halawt id-Dunya, #07, #19 & #28
From Muwashshah "Aya Daraha bil-Haqq", #05, #11 & #15
From Mawwal Zanjaran "Amana Ya Leil", #27, #54 & #68