Vaccinated global travellers will finally be able to come to Spain from June 7th

Update. The Spanish government on Saturday June 5th published a state bulletin confirming that it will modify the entry rules for vaccinated non-EU/Schengen citizens from June 7th.

What has Spain now confirmed?

Spain has modified the criteria for the temporary restriction of non-essential travel from third countries to the EU and Schengen countries.
The standout modification is that people who wish to travel to Spain from outside the EU/Schengen Zone can do so from June 7th if they have a vaccination certificate and have had their full vaccination treatment or last dose 14 days before travel. 
In essence, vaccinated people have been added to the list of non-EU/Schengen travellers who are exempt from the ban on non-essential travel to Spain, which up to now had been mainly for Spanish nationals and residents, students, several different categories of key workers and in some cases spouses and family members of Spanish/EU and those who can prove force majeure reasons. More information here

There are no changes to the list of non-EU countries which are exempt from Spain’s non-essential travel rule. People from Australia, Israel, New Zealand, Rwanda, Singapore, South Korea, Thailand, Hong Kong, Macao, China, the United Kingdom and Japan can continue coming to Spain for non-essential reasons such as holidays.
The difference for vaccinated travellers from countries that are not on the list is that they as specific people are now also exempt from the non-essential travel ban, as long as they can prove they’ve been vaccinated.

The Spanish government has published a second state bulletin which lays out the new conditions for travel to Spain regarding vaccination certificates, health passes and more. 
Certificates of diagnostic test of active COVID-19 infection with a negative result issued in the forty-eight hours prior to arrival in Spain will be accepted as valid.

The limitation of flights between the Federative Republic of Brazil and the Republic of South Africa and Spanish airports is maintained until June 30.
Flights from Brazil and South Africa may only land at Spanish airports if they are carrying Spanish nationals or residents, Andorran nationals, travelers transiting through Spain to a non-Schengen country, provided they will spend less than 24 hours in Spain and will not leave the transit area of the Spanish airport. Government flights, search and rescue services and non-commercial flights with a stop in the Spanish territory are exempt. Cargo, ferry, humanitarian, medical and emergency flights and aeronautic staff required for air transport are also exempt from the air travel ban.