So, as per usual, some distinctions are likely in order. I disagree with the one commentator who inserted "[public]" in one spot because he decided it's clearly the intent of the lawgiver. That's not generally an accurate approach to Canon Law. Those who write the law are very deliberate about every word or phrase and its placement, and the other source IMO makes the correct interpretation. Interpretation is usually with regard to circuмstances that come up that fall into a gray area not explicitly detatiled in the law, but not by inserting words into the law that are not actually there So, where that one source inserted the "[public]" is in front of the "Exsequalis", which refers to a Mass of Christian Burial proper, but the other source said that while some Rites may be held, not the actual Requiem Mass (thereby interpreting the law without that insertion).
Here's an important distinction:
1) Formal Funeral Mass
2) Ordinary Requiem Mass (not the formal as per above, often said by priests on Ferial days)
3) Having a Mass said for the intention of the repose of the individual's soul
I would read the law as simply indicating no on #1, the Formal Funeral Mass. That's where the departed individual is repeatedly named and there are specific Rites that lead to formal Christian Burial. Requiem Mass is somewhat more generic, and often one just commemorates the name of the departed in the Canon. Then there's any Mass with a simple Commemoration of the name.
Second set of distinctions is between ...
A) Private Mass (said by a priest without faithful in attendance) -- rare these days due to priest shortages
B) Public Mass but with Private Intention
I think that even #2 would be licit if the so-called "Private Mass" is allowed, meaning that even if the Mass per se is public, if the intention is kept private, that would be licit, IMO.
I personally think that if there's some reason to think there may have been extenuating circuмstances, #3, a Mass in which at least the intention is kept Private, would be permissible. Or with a Private Intention you could simply offer it to God conditionally where 1) you may apply the thinking of Padre Pio to request (retroactively) the salvation of the individual and/or 2) if the individual was saved, some relief from their Purgatory, or 3) for some other intention, before God, if neither of those two apply. So it's like a set of "conditional" intentions, not unlike you would conditonally confer some Sacraments.
Even with that last one you have to be careful since if there's no positive reason but just negative (wishful thinking) hope that the person may have been saved ... you don't want to create in your own mind any sense that ѕυιcιdє is not objectively grave sin that generally leaves little hope for salvation, and nearly always ends up with the loss of a soul.
#2 I think is borderline, since in a sense it's just a certain type of Mass with the intention directed toward ... the benefit of that individual. But, if there's any sense either with the one requesting it or the priest that it's closer to the Category of #1, I think it should be avoided.
In general, IMO, the best approach by far would be to request a Mass for your own private intentions, and then make the private intentions as outlined above.
That I believe is most in keeping with the mind of the Church, i.e. making it TRULY private.
There's also a third distinction here but I saved it for the end because it would be confusing the other distinction with the same terms, Public vs. Private. So, in Liturgical theology, Public refers to any prayer that's an officially sanctioned and/or prescribed Rite of the Church, even if the priest offers it competely alone. Mass and Divine Office are intrinsically public prayer of the Church, so in that sense there's no such thing as a "Private Mass", and by submitting an intention for a departed ѕυιcιdє, it's in a sense already encroaching into the public prayer of the Church, and the mind of the Church has been to not publicly prayer for departed ѕυιcιdєs. That's another reason that just requesting a Mass to be offered for a "Private Intention" would be the best course of action, and then you offer your intention to God, making it clear that it's conditional, as stated above. Normally if you ask a priest to offer a Mass, that's not typically a place for the generally long convoluted type of conditional intention, and that's why "Private Intention" suffices, and then you also don't wait for a rare opportunity when a priest might be travelling and have to offer Mass alone in a hotel room, but it could be offered at the main altar of your chapel with the priest announcing and/or publishing "Private Intention of [your name]".
To me that's the best of all worlds, keeping consistent with the mind of the Church, and yet giving you greater opportunity to get such a Mass offered, since I bet that only a small percentage of Masses are actually able to be offered as Private Masses.