If you already know one braid, it helps to see which others are related to it. A few small changes to the basic pattern can make a lot of variations without having to learn an entirely new sequence. You can also create your own versions.
I've created a symbolic short-hand to show at a glance how the instructions combine standard elements to create different patterns.
A basic symbol represents the braid:

or

Which can be joined with another worker's braid:

or
To that, the following might be added:

The outer edges are connected to form a tube.

A second color is brought to the surface at regular intervals.

An open edge is joined using a full twist.

Several repeats are performed before the pattern changes.
Lastly, 5.3 indicates that the braid requires two workers: the one on the left with 5 loops, and the one on the right with 3 loops.
Easiest Braids
The lowest loop of one hand is pulled through the top loop of the other hand, crossing over any other loops. Other than joining with a partner and playing with colors there's not a lot of design potential to explore. The instructions specify to take the loop reversed, but unreversed makes an interesting braid as well.
Twill-like Braids
The next three groups all share the same pattern - the lowest loop (the Key loop) on one hand is pulled through
all the loops on the opposite hand. The pattern is the same regardless of the number of loops used. Only a few of the many possible combinations appear in the manuscripts.
Round braids: The Key loop is always taken reversed. This connects the top and bottom layers on each edge, making the braid roundish.
Unconnected braids: The Key loop is always taken unreversed. This keeps the top and bottom sides of the loops from intersecting - two separate braids result. There's a unique potential here for forming the braids into tubes, with one tube completely covering the next, and for switching the colors between the braids.
Mixed braids: The Key loop is taken reversed by one hand, and unreversed by the other. This connects the layers along one side while leaving the other open, forming a braid that can be unfolded into a wider structure. This braid is interesting combined with the round and unconnected braids.
Tabby-like Braids
The basic pattern for these braids forms a tabby-like structure. They are presented in five and seven loop versions, but are similar enough to each other that certain combinations work for either one.
Five-loops: The Key loop is taken reversed on one hand, and unreversed on the other, forming a braid that can be unfolded into a wider structure. The manuscripts don't explore other options, but every variation that can be done with the twill-like braids can be done here, including making them round, unconnected, and joining with a partner's braid.
Seven-loops: The Key loop is taken reversed for a round braid, unreversed for two unconnected braids, or both to connect the braids along only one side. Additionally the seven-loop braids explore twisting the outermost loop to join open edges, allowing for long uninterupted stripes of color.
Serene's Oddities
(I'll classify these later)
Swivel Braids
Rather than being formed by multiple interlacements, this braid is held together by one element twisting around another in a regular pattern. It needs to rotate while being worked.
Exchange Braids
These braids are as simple as pulling one loop reversed through another loop. Loops are exchanged between the hands in different patterns, creating the variations seen here.
Bend Braids
A
bend is the heraldic term for a diagonal line. All these braids are formed by pulling each loop in sequence from one side through all the remaining loops to the opposite side.
Parallel bends: When two people work togther, the traveling loop passes through all the loops of both workers, forming one very large bend. The working direction can be changed at regular intervals, altering the direction of the diagonal.
Opposing bends: Two workers can join together opposing diagonals, producing what looks like a large chevron or "V" shape. Altering the direction of the diagonal at regular intervals makes "cross and eyelet" patterns.
Smooth-sided bends: The problem with braids that are formed along a diagonal couse is that there are long floats when that course is reversed. This braid is cleverly composed of triangular sections, which allow the direction of the diagonal stripe to change while keeping the sides of the braid straight.
Chevron Braids
A
chevron is the heraldic term for a "V" shape. They are really just smaller diagonal braids (one worked on each hand) joined in the middle and again between workers for the double-wide versions. With a few exceptions, all of the braids based on chevrons include both round and flat versions, for both one and two workers.
Repeating chevrons: This is the most straight-forward option, with no changes in diagonal or working direction.
Four repeat inversions: As with the "bend" type braids, the direction of the diagonals can be reversed to form diamonds, crosses and eyelets.
Eight repeat inversions: The diagonals can be reversed after any number of repeats, giving different effects. The braids for two workers are a glaring problem. They specify a color arrangement that would work for the "bend" braids above, but cannot work for "chevron" braids. Braids based on chevrons require symmetrical color placement across all hands.
Two repeat inversions: Reversing the diagonal after every two repeats means that only one color forms the stripes of the pattern. The dominant color can be exchanged as desired, with striking effect.
Mystery Braid
This braid is an oddity on many levels. It uses loops, but is structurally the same as a seven-element braid made without loops. It doesn't even live up to its name, being neither wide nor striped.