It is sunnah to sit between the sermons. it is stated in fatawa hindiyyah that if a person arrives at the time of khutbah then he may sit down in a squat like position (إحتباء-ihtiba': sitting on hips while knees bent and hands wrapped around his legs) if he likes, or may sit easy (chaar zanu baithna - with thighs on the ground), or whatever manner is easy for him* because khutbah is not an act of salah, or salah in essence. the above excerpt hints that there is no specific way of sitting between the two sermons, except that it is preferred to sit like in salah. However, Mufti Ahmed Yar Khan Naimi alihi rahmah states that some elders say, though he did not give any references, that one should fasten hands like qiyam in the first sermon, and put them on thighs like in tashahud in the second, it is hoped that he will get thawab of two raka'ah due to the fact that the khutbah of Friday is equivalent to two raka'h of zuhr prayer. ps: edited to fix a mistake in the translation of FH. * however he should not cause discomfort to others or sit in a manner which is considered against the adab of masjid and majlis
On several occasions, this Q has been posed on Noor TV/Mufti Akmal Q&A. As far as I can recall from the answers, it's recommended to assume an attentive pose (and to sit in the manner of tashahud is obviously the most attentive/humble way). To sit and change pose in the second part (as you have described) is merely mustahabb. This was not a specific/binding practice among salaf that has been emulated for generations. I suspect it is more a practice that is prevalent only among sunnis of subcontinent. I hardly used to see this practised (i.e. releasing hands in the 2nd part) during my early days (but then I grew up in places where most sunni Muslims were simple, not very deeni, ilmi).
I've heard it's been attributed to Ala Hazrat. Also, that Mujadid Alf Thani preferred to sit entirely as if in tashhud i.e. hands near knees.
Does anyone know where the practice of binding the hands during the first part of the khutbah, and then releasing them (as if in tashahud) during the second part is from?