It is unfortunate that the Wikipedia page promulgates the dubious distinction between register systems and contour systems. Mandarin Chinese has two tone registers (ergo the high-rising and low-rising tones); many Bantu languages have contrasts in level, rising and falling tones, but also don't employ downstep and have only two lexical registers. You may want to read Ken Pike's book Tone Languages: A Technique for Determining the Number and Type of Pitch Contrasts in a Language, with Studies in Tonemic Substitution and Fusion, which sets for the distinction based no what he knew at the time. But I don't deny that the terminology historically was used, and we just have to deal with it.
The concept of register does two things for tonal analysis. First, it is used to organize primitive levels. For example, Ewe has 4 tone levels (and has no contour tones), which we can label (from lowest to highest) L, M, H, R (R standing for 'raised H'). These tones phonologically organize so that L,M are 'a group', and H, R are 'a group'. There is a rule where M→R when between H,R tones. The analysis is that these tones basically organize into L versus H tones, but there is also a lower register vs. a higher register, so L in the lower register is called 'L' and L in the higher register is called 'H'; H in the lower register is called M and H in the higher register is called R.
The other thing that register does is define global shifts in reference pitch. In some languages, you can bounce back and forth between L and H tone and the Ls are more or less at the same level, and the Hs are more or less at the same level: L and H are computed relative to a single reference value. This is the case, for example, in Lingala (one guy's dialect), and Matumbi. But there are also languages like Shona where in an utterance with the sequence L H L H L H, every time you make the transition L H, you redefine that reference value, so numerically if we use 1 as the highest value, the pitch sequence of the above would come out something like 3 1 4 2 5 3 (this is probably the most common system in Bantu, with a caveat that the languages often also have 'downstep', q.v.). This phenomenon of automatic global lowering is known as 'downdrift'. The concept 'register' is involved based on the premise that H and L are defined relative to a reference value – the register – and the register can be lowered by rule. There is no principled limit to how many times pitch register can drop, though there are practical considerations on how low you can go (speakers plan ahead; speakers sometimes have to reset).
The aforementioned feature 'downdrift' is phonetically transparent, since it is triggered by an overt L; but many (maybe most) languages with this feature do not require an overt L, and then you can get lowering between phonologically identical tones. This is called 'downstep', and is traditionally marked with a raised exclamation mark (or the symbol ꜜ). It is actually not possible to tell by simple listening whether a language has three levels (H, M, L) or two levels plus downstep. The concept of downstep was thus introduced by Winston in a paper on the so-called mid tone of Efik. There is an analogous property of register raising which is attested in exactly one language, Acatlan Mixtec, where tones can recursively (and contrastively) increase register value (thus the basic tones go up, up, up and not down, down, down).
There is obviously some redundancy in this way of conceptualizing tones, because L is 'step down from H', but downstep is ‘step down’. Without knowledge of Acatlan Mixtec, we might say that 'basic tone' can involve both 'step up' (L to H) and 'step down' (H to L), but register only involves 'step down'. In fact, though, step up and step down are operations both at the register level and at the level of basic tone. There have been (distinct) theories by Keith Snider and Mary Clark which attempt to have a single raising vs. lowering operation. You can read this paper for an overview that focuses on African tone systems and points to some of the less-frequent patters regarding basic levels and downstep operations. The point is that there is a phonological distinction between lexical level, where a given syllable can, in isolation, have one of 2 levels in a 2-level language like Shona, 3 levels in a 3-level language like Yoruba, and so on, up to 6 attested levels in Chori.
I would say that the question of whether a language also has contour tones is synchronically orthogonal, although we know from comparative studies that languages like 5-level Kuteb (which does have rising and falling tones) and 6-level Chori developed these extra levels by flattening out earlier contour tones (however, that is not the only source of level-splits, which also arise because of properties of preceding consonants). There is probably a perceptual tradeoff that it is really hard to distinguish 5 levels of tone plus maintain all of the possible rises and falls that 5 tones allow, and thus the number of contours is usually fewer than the number you would predict by free combination of levels.
