The OF rubrics call for the consecrated elements to be shown to the people. They don't get more specific. John XXIII did not celebrate according to the OF.

Remember, rubrics of the EF evolved over the centuries when the laity generally did not receive sacramentally, so the elevation praxis developed with spiritual communion in mind (of course, if there was a rood screen or other veiling of the sanctuary, the people did not even see this...).

The other thing to remember is that the Eucharistic sacrifice is not only a re-presentation of Calvary, but of the whole Paschal Mystery: the Last Supper, Calvary, and Resurrection (especially as the manifestation thereof figured on Easter evening at Emmaus) - as well as Pentecost and a foretaste of the Wedding Supper of the Lamb in the new creation after the end of time. So it won't look like just one of those things, but point to all of them in some way.

In the EF there are special rubrics for the Holy Father celebrating Solemn Papal Mass, which direct him to elevate the Sacred Species in a semicircle to the Epistle side and a semicircle to the Gospel side. I don't know how or even whether this is regulated in the OF.

I don't think there are any official papal Mass rites any more. I think they are just adapting some of the older ones "as they go along" so to speak.

wrote Ben Yanke.

Similar things are done in the Anglican world, especially if a Westminster Abbey service will involve one or more HRHs, or the Bishop of Rome. When Pope Benedict visited the Abbey with Archbishop Rowan, it could have been just splendid if it were the normal Choral Evensong of the day, with psalms and lessons as appointed, perhaps with an extra-special anthem and some time for a homily from each. But it ended up an Evensong-ish hybrid, sort of a highlights from Evensong. Similar things happen for funerals, where something super special apparently must be created for TV.