The broadly-accepted imperative requirement is "vaccinated", as in "do not enter without a mask/etc unless you have been fully vaccinated". There is never a reference to "inoculated" (to wit vaccinated or had Covid and recovered).

Furthermore, there may not be "official documentation from your gov't and official health body". I was tested & 'treated', but my wife had exactly the same symptoms & recovery without bothering with doctors (seriously, it was really mild), so there is no documentation and no apparent process to procure such documentation.

"Any sufficiently advanced technology is indistinguishable from magic." Seems that, to most people, Covid & vaccines are magic: the disease is perceived little differently from evil spirits (sparing some and slaying others, with nothing corporeal to perceive & avoid), and the vaccine is a tangible talisman mysteriously able to ward off such spirits (by a mere prick from a costumed wizard wielding magic potions). This leaves most people, unable to really grasp the objective mechanism of vaccination (being a simulation of actual infection, to teach the body to fight it off), with the only objectively perceivable act they can grasp in the issue: vaccine injection. Hence many dismiss the near-spiritual notion of "I contracted it, survived, and am now immune - without intervention by wizards". Thus, to present it a bit over-the-top, recovered and vaccinated are not treated the same by those not professionally involved in medicine.