There's generally two types of prayers in Buddhism as far as I know:
Prayers of dedication are used to dedicate the merit of a practice to a particular being, groups of beings (usually all beings) , or some purpose.
Aspirational prayers are typically calling on the power of some Buddha or Bodhisattva to fulfill some wish that we have.
There are also special purpose prayers used for something specific.
For example, many Vajrayana sanghas will recite the seven branch prayer to request the guru to teach.
I don't think this qualifies as an aspirational prayer, but I may be wrong.
In my experience, generally we say prayers to dedicate our merit to all beings and say aspirational prayers that all beings may be enlightened.
As for the necessity of prayer... Well my tradition places a very strong emphasis on dedicating merit after each session.
Tons of aspirational prayers are used as well, but they're not as central.
Dedication here means dedicate your merit to someone or something.
For example, "I dedicate my merit to all sentient beings."
We do this to share the merit we accumulated with other beings so that they can have the same benefit without doing the work.
Yes, you're mostly right about aspiration.
Except you don't always wish for things yourself.
For example, "may all beings be happy and free of the causes of suffering ".
Well, the terms are more generalized than that.
Anything at all can be dedicated or aspired, not only happiness.
If I am assuming correctly, it seems like you're trying to translate my responses into your own framework.
I don't have any problem with that, but I will say that I'm not sure exactly what your framework is, so I cannot speak in that language.
I can only respond from my point of view.
dedication is kind of like "I'm happy, and I want to share my happiness with everyone else"
I think your understanding of dedication in general is correct, but that is not the correct way to view the dedication of merit done after one's practice.
Dedicating merit is not sharing your happiness with another being.
Karmic merit is understood as actually real, and we are sharing that merit with other beings.
I will say, though, merit is a rather complex topic.
For example, the accumulation of merit and wisdom is seen as one and the same as the removal of emotional and conceptual obscurations.
There is a lot to be known about merit, I'm sure.
I only really know those basics I just shared, but I remember there being a lot of detail on it in Chapter 4 of the Abhidharmakosa.
aspiration is "I want everyone else to be happy" ?
Again, aspiration can be more general than simply aspiring for happiness. One can aspire for anything.
Aspiration can be a general statement of aspiration like that, but more often one is actually calling upon a Buddha or Bodhisattva to fulfill one's wish.
(Yes, Buddhas and Bodhisattvas actually exist.)
The Buddhist sutras were originally passed down through oral tradition.
They were thus chanted, which is why all the repetition.
Sometimes they are still chanted, both to memorize sections of the canon and as a meditation, which I suppose really crosses the line into prayer, depending on your definition of either.
My experience with the zen chants are that they help to encourage practitioners along the path, they convey teachings, they help focus the mind, and can serve as a form of devotional prayer along the lines of what .
Many times the chant is in a foreign language and you are not even expected to understand the words - there are chants that are not sutras, or are fragments of sutras.
Other times you make a deliberate offering of merit or goodwill to all beings.
I personally think that prayer is just like a mantra.
For a prayer to be effective it has to be fairly long, more than several minutes.
When you chant in Buddhism, there is a certain unity and cessation of thought, just the mantra. I view the prayer the same way.
I saw some Christian Orthodox chanting/prayer, I am not sure what they call it, but ordained monks were all chanting in harmony. The same with Buddhist chants or mantras.
I really think this is what it comes down to, just praying and chanting any scripture that you like, long enough to attain this cessation of thought and unity with the chant that transcends to Buddha mind or something divine.
I don't like short prayers when people ask divinity for something, it's a bit narcissistic and fruitless.
This is just my opinion but essentially I do think that back in the day people used to pray with full passages and full sutras, rather than short prayers. I think you can enter a certain samadhi with prayer or a mantra chant, it doesn't matter.
Prayers as a part of Buddhist practice has been a part of Buddhism since forever.
Westerners may not like to hear to this, but Buddhists often pray for material things.
Go to any temple in East Asia before exams time and you can see many student praying for success.
People pray for the success of their business, for the health of their loved one, a promotion ...you get the picture.
This is the reality that I know in China, Japan and Thailand. I suspect it's the same in other countries.
Most believe that the Buddha will somehow 'grant' them their wishes , but there is no textual support for that.
In Nichiren Buddhism we also pray for whatever we want, especially at the early stages of practice .
This has several functions - first it allows you get acquainted with areas of your mind that you may overlook .
Ultimately though, as the practice deepens you find that what you "want" is less personal and more inclusive.
So the personal material aspect that was important in the beginning of the practice is used as an "expedient means" or a "hook" to create a steady practice and discover spiritual resources.