"Judge Rapporteur" is the term used in Luxembourg at the European Court of Justice and the General Court, and in Strasbourg at the European Court of Human Rights. Because of its brevity it may provide a more concise translation, but “judge rapporteur” is not common in the US or the UK where “writing/delivering/giving the opinion of the court” is preferred.

I believe “recording” should be avoided when translating "ponente", since judges “deliver” or “write” the opinions of a court, while non-judicial court personnel, usually a court clerk (“secreterio judicial”), record or enter judgments on the court record.

Here are examples of how “writing/delivering the opinion of the court” is used in both UK and US courts:

The Lord Justice General, writing the opinion of the court, said:. "In view of the difficulties which have arisen with the English prison authorities since ...
www.publications.parliament.uk/pa/ld199495/ldhansrd/vo94112...

Writing for the Court, Justice John Paul Stevens said: "It is true that we have repeatedly recognized the governmental interest in protecting children from ...
www.ee.surrey.ac.uk/Contrib/Edupage/1997/06/26-06-1997.html

Writing for the Court, Justice Sandra Day O’Connor found that the law school had a compelling interest in attaining a diverse student body and that, ...
www.cup.cam.ac.uk/us/catalogue/catalogue.asp?isbn=052175071...

Lord Reed, delivering the opinion of the court, said: "Where the summons has not called within the period specified by the rule, the automatic consequence ...
www.journalonline.co.uk/news/1001340.aspx

The Opinion of the Court was delivered by the Lord Justice General. In that Opinion, he recorded that in the course of the hearing of the petition a number ...
www.scotcourts.gov.uk/opinions/MISC117.html

L Phillips MR (delivering the opinion of the court) acknowledged that the difficulties the current “multiple publication” rule posed for Internet publishers ...
www.law.ed.ac.uk/ahrb/script-ed/elaw/loutchansky.asp

The case of these jurymen was reviewed on a writ of Habeas Corpus and Chief Justice Vaughn delivered the opinion of the court which established the Right of ...
www.ccguide.org.uk/law.php

The Lord Justice Clerk (Lord Cullen), delivering the opinion of the court, said (at 365B):. "In our view the plain meaning of subsection (3)(a) is to ...
www.scotcourts.gov.uk/opinions/2006HCJAC36.html

Giving the opinion of the court, Lord Justice Clerk Gill said: "On the information before us, we conclude that the Society is neither secret nor sinister ...
www.law.ed.ac.uk/sln/index.asp?page=218

Giving the opinion of the court, the Lord President said it was clear that the effect of the statutory provisions, when read together, is that the court ...
www.slab.org.uk/.../!SSL!/RoboInfo_Output/CHAPTER_19EXPENSE...