This is rather a knotty little group of verbs that have gone all over the place in Modern English, though they were very clearly and regularly distinguished in Old English.
Let’s start with the etymology, and then move on to current usage.
The words watch, (a)wake, (a)waken all share a common root. The Proto-Indo-European root was *u̯eǵ-, and its basic meaning approximately ‘to be strong/quick/lively/aroused/powerful’. It underlies for example Sanskrit वाजः vā́ jaḥ ‘power/force’, Latin vigil ‘watchful/awake’ (as in English vigil also), vigere ‘be lively’ (as in vigorous in English), velox ‘fast/lively’ (from *veg-lox; as in velocity in English).
In Proto-Germanic, the root regularly became *wek-, with the ablaut form *wak-. This latter form generally took over, because it was found in a very common construction with this root: the causative (PIE) *u̯oǵ-ei̯e- > (PG) *wak-ja- ‘make lively’ = ‘awaken [someone], wake [someone] up’. So for all practical purposes, the Germanic root was *wak-.
There were then at least three different verbs derived straight from this root present in Proto-Germanic (note: *-aną- is the reconstructed Proto-Germanic infinitive marker):
- A simple verb *wak-aną-, retaining the original meaning of the basic root, ‘be lively/awake’
- A nasal inchoative1 *wak-n-aną-, meaning ‘become awake’ = ‘wake up [by oneself]’
- The causative *wak-j-aną- mentioned above, meaning ‘make lively/awake’ = ‘wake [someone] up, awaken [someone]’
These were all quite regular, since both the causatives and inchoatives were regularly formed in Germanic by adding the *-j- and *-n- suffixes.
In Old English, they were also relatively distinct, although some parts of the paradigms had already started to coalesce. Most notably, both the old base verb and the old inchoative verb had become strong verbs in verbal class VI. This is original to the base verb (which is a primary, non-derived verb and therefore shows ablaut—the Proto-Indo-European equivalent of Germanic strong verbs), but not to the inchoative verb. The inchoative verb is a derived one, and it ought to have been weak; but it had fallen together with the base verb in the past tense.
This coalescence is very clear if you look at the Wiktionary paradigms for Old English wacan and Old English wæcnan: the past tense forms are the same, the others are distinct.
The causative *wak-j-aną-, on the other hand, had maintained its different conjugation throughout: it remained a weak verb of class II, Old English wacian.
So at a brief glance, if we give just the third person singular of the three verbs in Old English:
- Wacan: present wæceþ, past wōc
- Wæcnan: present wæcneþ, past wōc
- Wacian: present wacaþ, past wacode
In Middle English, when final syllables started being reduced and lost, the differences between these three paradigms started getting muddled. The Middle English infinitive forms are the ones you have found on Wiktionary yourself:
Once the forms started getting muddled, obviously the meanings started to overlap—it’s difficult, after all, to maintain a difference between ‘becoming [by one’s own accord] awake’ and ‘causing [someone] to be awake’ and ‘being awake’ when half of them have the same shape in various forms in the paradigm.
In current English, there are only two basic forms left: wake and waken.
Wake is, in form, basically from the old base verb waken < wacan. The modern verb waken, on the other hand, is from the inchoative waknen < wæcnan. Note that in both these words, it is the final -en that has been lost over time, and modern waken thus represents Middle English wakn, to which of course an extra little prop vowel was added, since /kn/ is not a valid way to end a syllable in English.
In Middle English (as in Old English), both waken and waknen generally had a strong past form, wook (in various spellings), which is what gives modern woke. The weak causative wakien had a weak past, waked(e). When the form of the causative was mostly lost (merging with the base verb once -en disappeared), the weak past was kept in some dialects, the strong past in others, and both in yet others.
On the other hand, the inchoative waknen developed a new weak past of its own, wakned(e) and stopped using wook like the base/causative verb. So there was a shift in what verbs shared what past tense forms:
- [base + inchoative] strong past vs. [causative] weak past >
[base + causative] strong or weak (or both) past vs. [inchoative] weak past
This is the reason why waken today has only a weak past form, wakened, while wake can have both strong woke2 and weak waked.
When the causative disappeared as a separate form, the rather unusual thing happened that its semantics were taken over not only by the base verb (with which it merged in form), but also by the inchoative verb. This means that both the base verb and the inchoative verb can now be either transitive or intransitive, which is a messy situation, but reality.
Waked vs. woke in current usage
In current English, woke is the standard past tense of wake, both transitive and intransitive, causative or not; waked is marked as nonstandard (dialectical) or archaic, and it’s nowhere near as common as woke.
An exception to this is when wake is not the inherited verb(s) discussed above, but rather a different verb derived (through normal zero-derivation) from a noun. Wake as a noun has at least two common meanings (etymologically unrelated): a vigil after the death of someone; and the trail of water or air left behind a vessel. When either of these nouns is used as a verb, the verb is a secondary, derived one, and such verbs are (nearly) always weak.
That means that the local community waked old Mrs. Smith last week after she passed away; you would never say that they *woke her. Similarly, you might possibly say that the ferry waked its way through the narrow strait; but certainly never in a million years that it *woke its way.
(Thanks to @supercat for mentioning this in the comments below.)
1 There’s quite a bit of discussion about what the exact usage of these nasal formations were in Common Germanic. They are often based on nouns describing people or jobs, and they then mean ‘do/behave as X does’, a basically causative or factitive meaning; thus, in Old English, we get læcnian ‘heal’ from lǣċe ‘doctor’. But they are also used more or less as inchoatives from adjectives, as blacken ‘become black’ from black.
2 As Jon mentions both in his answer and a comment here, woke, phonemically /woʊk/, is not quite regular from the Old English wóc. Some analogy with similar-sounding verbs like break, speak, and steal (which did sound the same earlier on, though they’re different now) took place at some point. Middle English wook or wooke or woke can be regular from wóc, assuming that they represent /wuk/, which is what the Old English form should give—but that’s not necessarily a given, considering the vagaries of Middle English spelling.