I have lived in England, Singapore and Australia where the 3 volume Divine Office is used. It is not referred to as the Liturgy of the hours over there, although it is the same thing using a different translation. The UK Divine Office uses the Grail translation of the psalms and includes the chant marks for chanting the Office. The US 4 volume LOH covers the same material but uses a different translation that is better suited to recitation than chanting.
A breviary is what we called one volume of the Office. For example, I need to go get my breviary - referring to current volume that we were using, or I need to mark my breviary (place markers where the next Hour of the Office was to be recited.
The UK version does not use a single prayer for midday, although this can be done by choosing one Hour from either Terce, Sext or None (before noon, midday or after noon).
Matins is another term for the Office of Readings and in addition to the psalms, there are two long readings as well. This Hour of the Office can be ‘anticipated’, that is, it can be prayed the evening before or any time on the actual day - for example, Thursdays Office of Readings (sometimes the term “Office of …” is used to refer to a particular hour of the Office - can be done on either Wednesday evening or any time on Thursday.
The Benedictines of the Ancient Observance use the Roman monastic diurnal which preceded Vat 2, and it is chanted in Latin, The page is divided into two columns with one side in Latin and the translation into English on the opposite side. They also continue to do the Hour of Prime, which was eliminated after Vat 2.
Any version of the psalms may be used while reciting the Liturgy of the Hours, so in my convent, we currently use different books with different translations for different Hours. We chant some of the Office, so we use psalms designed for chanting sometimes, but we also use different English translations for Morning, Evening and Midday Prayer. For Night Prayer (Compline), we use the 4 volume LOH set,and we also use the antiphons from this for the Propers.
Our Catholic liturgy is very rich and varied, and depending on which country you are in, whether you chant or not (Grail psalms are great for this), which translation you prefer, and which Order you belong to (if you are a religious), there is a wide variety of choice.
For a newbie in the US, I recommend either Christian Prayer or Shorter Christian prayer - both 1 volume books, but they don’t contain ALL of the same info in the 3 or 4 volume sets naturally - but enough to get started and to pray, especially for a busy person who lives in the world.
My favorite is the UK 3 volume set with the Grail translation but I am learning to love the variety of my community using the 4 volume LOH and the different books of psalm translations. It’s a very personal thing - but all of it is the official prayer of the Church, meant to sanctify the day by praying to God at different times throughout the day.