Q:

Naming the 13th chord on the leading tone of a major scale

In C major, there are seven diatonic 13th chords.
(Forgive me the ordering of extensions) Cmaj13, Dm13, Em11♭9♭13, Fmaj13♯11, G13, Am11♭13, and what? What is the name of the last one? The one built on B? Bø♭9♭13? Bm7♭5♭9♭13?
Also, has anyone ever tried naming 13th chords by their mode (C Ionian rather than Cmaj13, E Phrygian rather than Em11♭9♭13.

A:

B half-diminished 13th, or Bm♭13♭5.
Chord construction
When naming chords, it is best to keep it as simple as possible, to avoid confusion. That is: specify only the information necessary to unambiguously define the chord.
A 13th chord contains the following steps of the scale, in order 1, 3, 5, 7, 9, 11, 13. Any chord specified as a 13th chord will be assumed to contain all of the aforementioned tones, at least in theory. This distinguishes 13th chords from added 6th chords (e.g. C6, Cm6, C6/9); note that a C6 chord played with the A note an octave plus major 6th above the root is still a C6 chord not a C13 or Cmaj13 chord.
To explain the distinction, it's worth looking at the underlying principle of extended chord construction, namely the stacking of thirds. A major chord is two consecutive thirds stacked on one another, the first being major. A seventh chord stacks another third on top of that and so on.
To illustrate this, let's have a look at the G dominant extended chords:
X:1
K:C
L:1/4
"G"[GBd]4 "G7"[GBdf]4 "G9"[GBdfa]4 "G11"[GBdfac']4 "G13"[GBdfac'e']4

Given that the actual intervals may be major, minor, augmented or diminished, the chord name will need to contain some additional information to allow us to distinguish between chords built on the various steps of the major scale. The relevant distinctions are as follows:

Whether the chord is major or minor (does the chord contain a major or minor third),
In the case of major chords, is the chord major or dominant (does the chord contain a major or minor 7th; Cmaj7 vs. G7).
In the case of minor chords, is the chord minor or half-diminished (does the chord contain a perfect or diminished 5th; Dm7 vs. Bm7♭5).
The upper tone (in this case, the 13th can be major or minor; Dm13 vs. Am♭13).
Any important colour tones (♭9 for chords built on the third step of the scale, ♯11 for chords built on the fourth step).

We know that chords built on the 7th step are half-diminished (♭5) and we specify that the 13th is to be minor (♭13), avoiding any confusion between this and an m13 chord with an altered fifth. The intermediate extended tones (7, 9, 11) are assumed to be standard.
Voicings in practice
That doesn't address the issue of voicings and, in practice, one seldom uses all the chord tones implied by the chord name (either because of technical limitations to the number of voices, or to avoid clashing tones). To maintain the sound of a 13th chord, one will typically voice at least the root, 3rd, 7th and 13th (the inclusion of the 7th distinguishes C13 from C6, for example), plus the colour tone (♭5, ♭9, ♯11) where necessary.
To illustrate this in practice, let's try a four-part cadence using Bm♭13♭5. The key is C major and we know that half-diminished chords, being built on the leading tone, have a dominant function and can be used to create perfect cadences.
Now, Bm♭13♭5 is a seven-note chord (1, ♭3, ♭5, ♭7, ♭9, 11, ♭13) and we have four voices we can use, so we need to lose some parts of the chord. Normally, the 5th is the first to go, but in this case it's an important colour tone, so we must keep it. We'll also want the leading tone root and the ♭13, naturally. This leaves us with one voice, that we'll use for the ♭7, to highlight the fact that we're dealing with a 13th chord, as opposed to an added 6:
X:1
K:C
L:1/4
"Bm♭13♭5"[B,FAg]4

Now for the resolution. In a perfect cadence you'll pretty much always resolve the leading tone up a semitone to the root of the tonic. Similarly, the ♭5 of half-diminished chords wants to resolve a semitone down to the 3 of the tonic. The closest resolution for the ♭7 (A) is a tone down to the 5 of the tonic (G).
This leaves us with ♭13 (G). It's a shared tone between the two chords, so we could keep it, but there's a better way. The note is the root of the dominant, so we can make the standard leap a fifth down, just like we were resolving dominant to tonic in the root position:
X:1
K:C
L:1/4
"Bm♭13♭5"[B,FAg]2 "C"[CEGc]2

There we go! Try it on a piano to see just how well it resolves. The attentive reader will spot that Bm♭13♭5 looks suspiciously like a first-inversion G9 (no fifth), but if we were to do a root position G9 to C cadence, it would - arguably - sound a bit weaker:
X:1
K:C
L:1/4
"G9"[GBfa]2 "C"[Cceg]2

The reason for this is that in this case, the tonic of the root is doubled in the bass and tenor, whilst in the previous example it was doubled between the bass and soprano, which is the strongest form of a chord possible in four-part writing.
Personally, I have never encountered naming chords after modes of the major scale. I can see the elegance, but I would find it difficult to map the connection on the fly - at least without prior practice.

