We think in symbols, so our understanding is limited by what these symbols can represent. The more accurately our grammar and vocabulary allow us to model the world, the better results we can get.
Alfred Korzybski, in Science and Sanity , said that modern science started making progress when we changed our vocabulary and grammar. To the Alchemists, for example, "fire" was an object because the symbol they used for it was a noun. 
So they wasted lifetimes trying to isolate its basic unit, the "phlogiston".
Seeing fire as a process, by using the verb "oxidizing", enabled us to create plastics and isolate amino acids.
A specialized vocabulary can be words. It can be math, as in modern physics. It can be pictures, graphs or charts. It can even be a story.
Word problems are an elementary type of story that makes math easier to understand. A man has two donuts. Then he buys two more. How many donuts does he have? Children learn this pattern, trust it, and apply it to other things. Whether there actually is a man buying an extra two donuts at this very moment is entirely beside the point!
This is how the description of Amida Buddha and the land of Sukhavati works. It teaches us patterns important to Buddhism. If we recognize these patterns and see them in everyday life, that is "relying on Amida".
When the Sukhavati ("Pure-Land") practices were brought to North America, the only English religious terms available were Christian, and the Buddhist missionaries didn't understand how different the two groups were. This led to misunderstandings. For example, phrasing the Teachings this way leads many people to think that Sukhavati, instead of being a place where we can get over the graspings that lead to suffering, is a heaven where all these graspings are fulfilled.
Those raised Buddhist may learn otherwise. But newcomers, as well as more of each new generation, often fall into this trap of the script of Christianity acted out by a cast from Buddhism. We don't learn the patterns crucial to escaping from suffering that way.
The account of Amida Buddha, on the other hand, teaches us the pattern of Buddhism. We live in this world of suffering, but relying on Amida causes us to be reborn in the Pure Realm of Sukhavati. Once familiar with this pattern, we more easily perceive how our minds, by grasping preconceived ideas, lead us to disappointment through unrealistic expectations, but relying on what's outside our preconceptions, seeing things without judging them, shows us reality with nothing out of place.
Chinese Sukhavati tradition speaks of three requirements for Enlightenment, usually translated into English as faith, vows. and practice.
"Faith" is one of those words taken from Christianity. In Buddhism, it refers to the confidence that the Teachings are correct and effective.
Once you have this confidence, you will develop a strong determination to overcome the problems of unenlightened life. This determination is your vow, whether formally stated or not.
In our branch of Buddhism, practice is the use of the Sukhavati symbolism, most commonly by reciting Amida's name.
Buddha taught "anatman", non-self, that the ideas we are attached to, especially those about a "self", are a misinterpretation fueled by desires and fears. Since there is no "self", the difference between "self-power" and "other-power" is only a matter of teaching technique.
Buddhism points to the direct consciousness that has to be there, or we couldn't warp it with our hopes, fears, and habits of thought. When we pass a bulletin board and a notice about our hobby "catches our eye", some part of our mind had to read all those notices and reject the others we weren't interested in. Imagine having all that awareness and wasting it!
The so-called "self-power" schools then say "This unfiltered awareness is your Buddha-nature, your 'original face'. This tangle of delusions that lead to suffering is not."
Sukhavati Buddhism, though, says, "Since we use the term 'self' to refer to this unsatisfying set of preconceptions and habits, then you obviously have to rely on the unconditioned awareness outside all that. Let''s call it Amida."
Master Chu-hung said "Thus they awaken to the Amitabha of inherent nature and comprehend the Pure Land of mind-only" (from Pure Land, Pure Mind).
Bassui Tokusho Zenji said "This [kind of spiritual awakening] is called 'return to the origen' or 'rebirth in paradise'." (Dharma Talks on One Mind)
Self-power and other-power goals are not different, only their vocabularies are. But you see what a difference that made to the Alchemists.
"In this context, remembering Buddha, Buddha-remembrance, means remembering our own true nature, the capacity for lucid wisdom and selfless compassion that is our birthright." (from Mind-Seal of the Buddhas)
Understanding that, we recite the Buddha's name like an amnesiac who, feeling an inexplicable sense of hollowness and loss, finds an ID card in his wallet. He reads the name on it and repeats it to himself, tasting it.
And slowly, within himself, he feels something stirring....