There are others here more knowledgeable in this area, but from what I was taught in pre VII parochial school and as an altar server back then, and also from what I've read, the variations of the pre Vatican Council II liturgy were:
The Solemn Mass, also called a Solemn High Mass, which is a sung Mass offered by a priest assisted by a sub-deacon and deacon (or, usually, other priests filling the roles of the deacon and sub-deacon). If it is the principal Mass of the day in a parish, the Asperges rite (sprinkling with Holy Water) is used. Incense is used at the beginning, at the Gospel, the Offertory, and the Consecration. This was originally considered the normative manner for celebrating the Mass.
The Low Mass is a recited Mass, with a minimum of ceremony, and no incense. Its original purpose was for a priest to be able to celebrate a private Mass for his intentions, when he wasn't scheduled to celebrate a public Mass, which would have been a Solemn Mass. This was necessitated because the West did not retain (I'm not sure of the reasons) the tradition of concelebration, as did the Eastern Church. Concelebration has been restored in the Missal of Paul VI. From what I've read, the Low Mass was never intended to be a public liturgy, the tradition being that the Liturgy should be sung, but it eventually morphed into a public liturgy (perhaps that is an
organic development, but not a good one, I wouldn't think.
A Sung Mass, or
Missa Cantata was a Mass sung by a priest without the assistance of a deacon and sub-deacon. Incense was not used, as the rubric for it's use involved the deacon and sub-deacon. I'm unsure if the Asperges rite was originally used with a
Missa Cantata.
A High Mass is a more modern format (I almost want to say 19th to early 20th century, but I can't find a reference ~ I'm thinking it may have been introduced by St. Pius X, as he made many changes to the rubrics of the Missal) and is a
Missa Cantata with incense (handled by an altar server), and the Asperges rite is used at the principal Mass of the day.
Taking the liberty of offering an editorial observation (or comment

), in a former town of residence my neighborhood included a small Greek Orthodox Mission parish. Through some circumstances I became acquainted with the priest and presvytera who traveled 350 miles one way twice a month to serve the mission. As I am an
Ordinary Form (or Novus Ordo, if you prefer) Roman Catholic, and the Church discipline which I am under and recognize permits me to attend an Orthodox Liturgy on occasion, for appropriate reasons, I use to visit a few times a year the St. Nectarios Mission, as I loved their Liturgy. Without choir, or even altar servers, and often with only 6 - 12 people in the congregation, with Presvytera Sophorina serving as cantor (and all of the few people participating), Fr. Anthony would offer a
sung Divine Liturgy in three languages (English, Greek, and Slavonic), with incense at all of the indicated places, in about 1 hour and 15 minutes, including a sermon. After that experience, I've never understood, or accepted, the idea that the Low Mass should ever be a public Liturgy, and, if the Latin Rite would not provide for concelebration (which the East does, the
Novus Ordo does, but the pre VII understanding of current tradition does not at the moment), then the Low Mass should serve it's original function in tradition of being a priest's private Mass. I don't see any reason or excuse as to why EVERY public Latin Rite Mass shouldn't be sung, with incense. If the Greeks can do it ..., just saying ... but that's just my personal rant

.