Is Morse Code binary, ternary or quinary?
This answer isn't as long as it looks; this site just puts a lot of spacing between list items! Update: Actually it's getting pretty long...

Morse Code isn't "officially" binary, ternary, quaternary, quinary, or even 57-ary (if I count correctly). Arguing about which one it is without context is not productive. It is up to you to define which of those five it is based on your application and the encoding requirements for your specific situation. To that end, there are pros and cons to each symbol set size, which you should consider for your situation before making a decision.

For the basis, M.1677, the international Morse Code standard, section 2, defines timings as: 


  2 Spacing and length of the signals
  2.1 A dash is equal to three dots.
  2.2 The space between the signals forming the same letter is equal to one dot.
  2.3 The space between two letters is equal to three dots.
  2.4 The space between two words is equal to seven dots. 


Note that that defines timings only, e.g. a dash does not mean the same thing as three dots, it is simply the same length. The base unit of time is a dot.



Binary

We can choose binary if we'd like. It is possible to represent Morse Code with "strict binary", as you call it -- that author's approach was to assign the digits to "dot" and "dash". That approach does not work and I do not think the author has a firm grasp of Morse code. But there is a different approach that works. If we choose binary, we essentially represent whether the signal is high or low with each symbol occupying a single dot-lengthed time slice. Note that this is also the approach described on the List of Binary Codes page you cited.

Other binary codings can be chosen as well (e.g. Bob Jarvis' answer) at the expense of increases in difficulty decoding back to audible Morse Code (Bob's encoding cannot be decoded statelessly, for example, the tradeoff is it provides shorter encoded sequences).

Symbols:


Low, off, etc. (0)
High, on, etc. (1)


Encoding Examples:


Signal separator: 0
Letter separator: 000
Word separator: 0000000
Three dots: 10101
Three dashes: 11101110111
The word "son": 101010001110111011100011101 
The phrase "is a": 10100010101000000010111


Pros:


Minimal symbol set size
Easily managed with typical digital logic
Easy to decode into audible Morse Code (presuming no validation is done); stateless decoding and does not require storing information about the timing of symbols (every symbol is the same length).


Cons:


Long encodings.
Many opportunities for invalid encodings.
Individual symbol meanings (0, 1) depend on context.




Ternary

With this choice, we define letter and word spacings as rests, and signal spacings are implicit (although you could choose to require dot length rest symbols between signals, if it suits you better).

Symbols:


Dot-length rest (0)
Dot (1)
Dash (2)


Encoding Examples:


Letter separator: 000
Word separator: 0000000
Three dots: 111
Three dashes: 222
The word "son": 11100022200021
The phrase "is a": 11000111000000012


Pros:


Small symbol set size.
Signal separator no longer needed.
Shorter encoding lengths than binary.
Removes some invalid encodings (e.g. 0110 with the binary choice no longer possible).


Cons:


Still lengthy encodings.
Still many possibilities for invalid encodings, in particular, the rests.
Cannot be decoded statelessly† unless you choose to place explicit dot length rests between signals.
Signal separators are implicit‡ unless you choose to place explicit dot length rests between signals.
Some individual symbol meanings (0) depend on context.




Quaternary

Symbols:


Letter separator (0)
Word separator (1)
Dot (2)
Dash (3)


Encoding Examples:


Three dots: 222
Three dashes: 333
The word "son": 2220333032
The phrase "is a": 220222123


Pros:


Very short encoded length.
Possibility of invalid encodings is reduced.
Symbols are semantically accurate.
Arguably, more human-readable.


Cons:


Large symbol set size.
Cannot be decoded statelessly.†
Signal separators are implicit.‡




Quinary

A direct interpretation of Wikipedia's list. Note this is very similar to the binary choice except the various valid combination of binary 0's or 1's are given their own symbols.

Symbols:


Signal separator (0)
Letter separator (1)
Word separator (2)
Dot (3)
Dash (4)


Encoding examples:


Three dots: 30303
Three dashes: 40404
The word "son": 303031404041403
The phrase "is a": 3031303032304


Pros (compared to quaternary):


Can be decoded statelessly.
The only symbol set size that unambiguously, directly and concisely represents the five objects mentioned in section 2 of the standard: "dot" (to which all timings are relative to), "dash", "space between the signals", "space between two letters", "space between two words".


Cons (compared to quaternary):


Higher possibility for invalid encodings.
Increased encoding lengths without adding extra information (tradeoff is decoding ease).
Not as easy to decode as the binary choice, in that each symbol is a different length in time and this association must be maintained somewhere.




57-ary:

Included for completeness.

Symbols:


26 letters
10 numbers
20 punctuation marks and miscellaneous signs
Word separator (space)


Encoding Examples:


The word "son": son
The phrase "is a": is a


Pros:


Shortest encoding
Human readable
Invalid encodings are impossible.


Cons:


Largest symbol set
Difficult to decode into Morse Code; necessary to maintain tables of Morse Code output sequences for each symbol and necessary to detect word boundaries and adjacent letters to determine where to insert various rests.
Impossible to expand beyond the Morse Code specification without adding new symbols (tradeoff for being immune to invalid encodings).
Signal and letter separators are implicit.‡




Termination

For storage as a symbol stream on a device, if you need to do that, it's worth noting that M.1677 defines message and transmission termination sequences. From Part II:


  5 Every telegram shall be terminated with a cross signal (. – . – .)
  6 The end of the transmission shall be indicated by the cross signal (. – . – .) followed by the invitation to transmit signal K (– . –)
  7 The end of work shall be indicated by the station that transmitted the last telegram. The correct indication is the end of work signal (. . . – . –)


Unfortunately, the standard is silent on the ability to distinguish e.g. the string "+K" from the sequence (end-of-telegram, invitation-to-transmit). However, end-of-work is unique. If you are storing morse code, then, you will have to make a choice as to how you want to indicate the end of a stored sequence.

If you are storing an entire conversation, for example, and are using the binary symbol set, you could reasonably consider end-of-work to be the end of sequence indicator, represented as the unique bit sequence 101010111010111. You need to consider the type of data you are storing (e.g. message fragments, entire messages, entire conversations) as well as the facilities offered to you by your storage medium (e.g. ability to prefix with message length) before deciding. I suppose you could go so far as to define your own end of sequence indicator. In any case these are all matters of underlying storage and have too many possibilities to list here. The choice is really up to you based on whatever is appropriate for your situation.



† Cannot be decoded statelessly: These encodings require some state to be maintained in order to decode into audible Morse Code. They all require "If preceding symbol was a dot/dash and current symbol is a dot/dash then insert signal separator". While you could try and "cheat" and automatically append a signal separator to all symbols while reducing word separator to 5 rests and letter separator to 1, M.1677 afaict does not explicitly prohibit consecutive word separators, which breaks this kludge.

‡ Signal separators are implicit: This is directly related to the above note but more semantically oriented. These encodings have implicit signal separators in their notation. That is "zzz" is intuitively three separate z's, the boundaries are "piggybacked" on whatever it is that separates symbols in the encoding itself. Consider the description "three dots in a row"; intuitively we take this to mean three separate dots, but if separators weren't implied this could be taken to be indistinguishable from a dash. Only the Wikipedia-style binary and quinary (or ternary with the explicit signal separator option) representations remove all ambiguity and "shortcuts" here, which can be of great value depending on the situation. It is for this reason that there is a strong argument for the quinary choice.



TL;DR:

As you can see, there are advantages and disadvantages to each, and it's up to you to decide how you want to treat it based on your requirements. There is no right answer. You could attempt to argue that one is more appropriate than the other in general, but these arguments would boil down to pedantic preferences, as without context, there is no correct answer.

Now, all that said, in my own opinion, the quinary choice is the representation that feels the most "right" to me for Morse Code itself (originally I had favored quaternary). It is the only choice that provides all of the following:


Context-free symbols (a symbol on its own contains all information about its meaning without examining adjacent symbols).
Representation of all primitives defined in M.1667.
Separators (signal, letter, word) are not implied by notation or underlying encoding (this is important, if I wrote { dot, dot, dash }, those ,'s are significant and easy to overlook).


However, there are strong arguments for all. The arguments related to encoding are also applicable to the treatment of Morse itself. There is no 100% concrete answer, but it can certainly make a good topic for debate with your friends at the bar.

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