I think the tendency for punabhava and patisandhi to be regarded synonymously as rebirth, in English, contributes to the broadening of the spectrum of disparate notions and views that fall under the banner of "rebirth".
To me, as I understand it, punabbhava has nothing to do with transmigration, whereas patisandhi unambiguously does.
"Do not force others, including children, by any means whatsoever, to adopt your views, whether by authority, threat, money, propaganda, or even education." - Ven. Thich Nhat Hanh
"The uprooting of identity is seen by the noble ones as pleasurable; but this contradicts what the whole world sees." (Snp 3.12)
"To argue with a person who has renounced the use of reason is like administering medicine to the dead" - Thomas Paine