The situation you are interested in is not explicitly mentioned in Laws of the Game or specifically in Disciplinary Code of Royal Spanish Football Federation (PDF in Spanish, page 56).
Showing a direct red card after showing a red card for two accumulated yellow cards is redundant. Because what is more important, as you have already guessed, is what the referee wrote in the post-match protocol.
And here comes the most entertaining part: what has Ricardo de Burgos Bengoetxea mentioned in his protocol? Below you can see the original protocol, published by MARCA:
Which states:
Cautions:
79th minute. Player #7 Cristiano Ronaldo was cautioned for the following reason: removing off the shirt during goal celebration.
82th minute. Player #7 Cristiano Ronaldo was cautioned for the following reason: foul simulation.
Dismissals:
82th minute. Player #7 Cristiano Ronaldo has been sent off for the following reason: second yellow card.
Other incidents:
Once shown the red card Cristiano Ronaldo pushed me slightly in a sign of disagreement.
As a result of this incident, Cristiano Ronaldo was banned for five games. So, this decision has been made because the referee mentioned it in the protocol and regardless whether Ronaldo received direct red card or not.