Is it true that there are 5 principal types of meditations?

As stated by Sankha, there are only two types of meditation - concentration (samatha) and insight (vipassana). Even these two need not be mutually exclusive.

For concentration meditation, you can have various objects to concentrate on, like the breath (anapanasati), or loving kindness (metta) or unattractiveness (asubha) or any of the ten kasinas. The objective is to cultivate the jhana states.

For insight meditation, breath (anapanasati) is generally used, but the focus here is to observe the nature of reality based on the four foundations of mindfulness. So, even here, it can be on the body, feelings, thoughts or the different mental objects or dhammas, but without an intentional choice of an object of concentration.

Both are not mutually exclusively. For e.g. if you have a camera, you can zoom in and change other settings on your subject of focus, in order to get the best, sharpest and clearest photo of the chosen subject. That's like samatha.

But if you move your camera around to find good subjects to photograph, then that's like vipassana. The four foundations of mindfulness show you what subjects are good to observe, just like in photography you have landscape, animals, portraits etc. Eventually you need to use both techniques to get the best photo of the best subject.

It makes sense to me that learning how to focus your camera first (samatha), is preferred, before learning what kind of subjects to photograph (vipassana). If you try to take photos of your subjects before you learn how to focus your camera, your photos will turn out blurry or too dark or too bright etc (obstruction by the five hindrances). But in the first phase of learning how to focus your camera, you might focus on unimportant subjects like your toes (kasinas).

Another analogy is physics and mathematics. You want to study physics (vipassana) to learn about the nature of reality. You may be very eager to do so. But it would be unpleasant and difficult to learn physics in detail, before mastering math (samatha) first. You need to study math and master it to a certain degree, then you can use it to study physics smoothly and pleasantly.

In the first phase of studying math, you might use it to calculate unimportant things (kasinas) like using algebraic terms like x and y without having any specific meaning, for the sake of performing integration or differentiation on them. When you use it to study physics, then you can differentiate velocity v over time t, to get acceleration a.

I have found the reference to the 4 of the 5 kinds of contemplation mentioned in the question. They are taken from Visuddhimagga sanna(or sanne??). To quote -

Metta in the Buddha's teaching finds its place as the first of four kinds of contemplation designed to develop a sound pacific relationship to other living beings. The four are: metta, which will be rendered henceforward by "loving-kindness," karuna, which is "compassion" or "pity," mudita which is "gladness at others' success," and upekkha, which is "onlooking equanimity." These four are called Divine Abidings (brahma-vihara), perhaps because whoever can maintain any one of them in being for even a moment has lived for that moment as do the Highest Gods (the Brahma Deva).

To quote further from the document:

But first, before coming to these discourses, some details from the meditation manual, the Visuddhimagga or "Path of Purification," will not be out of place.

Metta (loving-kindness) is defined as follows: "Loving-kindness has the mode of friendliness for its characteristic. Its natural function is to promote friendliness. It is manifested as the disappearance of ill-will. Its footing is seeing with kindness. When it succeeds it eliminates ill-will. When it fails it degenerates into selfish affectionate desire."

The Visuddhimagga recommends going to some quiet place, where one can sit down in a comfortable position. Then, before starting the actual meditation, it is helpful to consider the dangers in hate and the benefits offered by forbearance: for it is a purpose of this meditation to displace hate by forbearance, and besides, one cannot avoid dangers one has not come to see or cultivate benefits one does not yet know.

Buddha says the following about the four sublime states:

I. Here, monks, a disciple dwells pervading one direction with his heart filled with loving-kindness, likewise the second, the third, and the fourth direction; so above, below and around; he dwells pervading the entire world everywhere and equally with his heart filled with loving-kindness, abundant, grown great, measureless, free from enmity and free from distress.

II. Here, monks, a disciple dwells pervading one direction with his heart filled with compassion, likewise the second, the third and the fourth direction; so above, below and around; he dwells pervading the entire world everywhere and equally with his heart filled with compassion, abundant, grown great, measureless, free from enmity and free from distress.

III. Here, monks, a disciple dwells pervading one direction with his heart filled with sympathetic joy, likewise the second, the third and the fourth direction; so above, below and around; he dwells pervading the entire world everywhere and equally with his heart filled with sympathetic joy, abundant, grown great, measureless, free from enmity and free from distress.

IV. Here, monks, a disciple dwells pervading one direction with his heart filled with equanimity, likewise the second, the third and the fourth direction; so above, below and around; he dwells pervading the entire world everywhere and equally with his heart filled with equanimity, abundant, grown great, measureless, free from enmity and free from distress.

The fifth one (Ashubh Bhawana contemplation) can be found here.

There are only 2 types of meditations.

1. Samatha(Concentration)
2. Vipassana(Insight)

All those 5 meditations you have mentioned and many more come under Samatha meditation. More on Samatha vs Vipassana...