PDF versions of the pentagram rituals:

queenofheaven418.com/lib_goldendawn.htm

The instruction state that the experience of the elements is the purpose of the pentagram rituals, with little more exposition of the intended outcome.

I have found three different pentagram methods for invoking and banishing single elements in the available literature. I have no idea of the cultural backgrounds of each method, and I don't think it matters which may be "right" or "original" or otherwise. What matters is how each method works, and how each method differs from the other two methods.

For want of better names, I call these three methods
1: the God Name Method.
2: the Pentagram Method.
3: the Chinese Elemental Method.

I will describe the three methods and my experience on their effects.

1. The God Name method.

Invoking this way always and for all four elements draws the pentagram from the Spirit Element to the Earth Element. Banishing this way always and for all four elements draws the pentagram from the Earth to the Spirit Element. What changes is the God Names vibrated:

Earth: AGLA.
Water: Eheieh.
Fire: ADNI.
Air: JHVH.

The God Name is used in each of the four directions for the Lesser Rituals of the Pentagram.

The effect of this form in my experience is to evoke the element in the very pure energetic form compared to the other methods. The risk is it can lead to imbalance very quickly. The opportunity is that this form is very powerful. I recommend using this form for times when the pure element is required on its own. I compare using this method to taking valium instead of valerian: valium is powerful and pure, but without the supporting and synergetic nutrients in valerian it can easily become addictive or numbing.

2. The Pentagram Method.

I have posted an image in this tribe's portfolio of all the forms of this method. It is quite simple visually, but complicated to describe in words. Simply put, the pentagram method invokes and banishes the first four elements (earth air fire and water) by moving from or to the element prior clockwise to the element desired; then the spirit element is evoked as ying or yang (or Sandalphon and Metatron if you like) by drawing from the denser elements (water and earth) to and from the lighter elements (fire and air.)

In other words, it is a bit complicated and confusing. But it works and works really well too.

In my experience this method invokes the element in a wholistic fashion. Because it uses all the god-names, it evokes the element in a very grounded way and is useful for daily practice - ie, invoking in the morning and banishing in the evening.

What I mean by wholistic can best be provided by an example. Say I use the pentagram method to invoke elemental air. By drawing the air pentagram with the four god names each time, I invoke the Air of Air (JHVH), FIre of Air (ADNI), Water of Air (Eheieh), and Earth of Air (AGLA). Thus the element is not isolated from its companions, and instead the airiness of the overall field is emphasised. Put another way, the pentagram method of working with the elements is best for everday living, because it brings out the elemental principles in things.

In my opinion this is the safest daily method to use.

3. The Chinese Method.

The Chinese method is about evoking a state of flow instead of a definite condition. It is considerably more subtle than the other two methods, and I think more powerful potentially as a long-lasting current. Using this method sets up a subtle current which can then be integrated into other magickal work.

The key concept is that each element either creates or destroys an adjacent element. Air, for example, is consumed by fire and enhanced by the evaporation of water - consequently, invoking air using this method involves starting at the water element and moving to the air, and banishing air involves starting at the fire element and moving to the air.

Meditation on the five elements reveals how each element is destroyed and created by the adjacent elements, and reveals which pentagrams to use to evoke the element.

The Pentagram Ritual using the Chinese Method is a creative and destructive act. Nothing is conserved; everything flows and changes. As such it is less "intrusive" than the other methods, but in my opinion more powerful because it engages the existing flow of elements directly. It can be used to alter hormonal and metabolic processes. It is especially useful for healing specific organs and for engaging the energy systems of the meridians. It is also an interesting adjunct to physical practices like martial arts, yoga and chi gung.

Have you checked out Liber O?
The vital information you feel is lacking, is due to the need of the student to work on it herself. I have found that many of the adepts, including Crowley will omit and mislead on purpose, so that you ask questions, such as you are now, and those whose time to do the Great Work has dawned on them, will work on the rituals for themselves. So, it is fantastic that you are asking these questions, and a clear sign you are being called homeward. The reason for the misinformation is that it is way more fulfilling to figure it out on your own, and ergo make the rituals your own, than if someone just handed them to you.
"Let the fine be tried in intellect..."
Each pentagram ritual has a specific use. One of the great workings of the magician (see atu 1) is to properly order the elements. All the elements are indeed reflected in ourselves and out of balance until we reach Knowledge and Conversation of our HGA. Each element is attributed to different aspects of our consciousness.
Earth-Survival instincts
Water-Emotions
Air-Intellect
Fire-Belief systems (such as religion, spirituality, philosophy)
The LBRP serves to banish in the way that it also invokes. It banishes, temporarily, the unbalanced elements (as Arun stated, earth being a solidification of all the elements) to create a cleaner vessel for the fifth element, spirit, to enter into our being, and be expressed. Ultimately, if you do, with fervor (meaning non habitual and passionate etc) the LBRP every day of your life eventually you will attain K&C. It will just take a while, like little drops of water into a bucket to fill it.