As far as I'm aware, there isn't a full 9th level prepared psychic caster. Obviously, we already have the psychic who is a spontaneous INT based caster. I believe there are archetypes that partially accomplish this.
But if there was to be a prepared psychic caster, what role would they fill? What ability score would their spellcasting key off of? Would they share the psychic list or have their own?
The only existing option for a prepared psychic Spellcaster of any kind is to play a Blood Arcanist with the Psychic bloodline. All writte psychic spellcasting classes and archetypes that convert prepared spellcasters to psychic are spontaneous. The Blood Arcanist with the Psychic bloodline is the only option out there that remains prepared (well simi-prepared anyways)... and it just so happens to be a 9th level caster
Class: Prophet, 9th level prepared psychic caster
Alignment: Any.
Hit Die: d6.
Starting Wealth: 2d6 × 10 gp (average 70 gp.) In addition, each character begins play with an outfit worth 10 gp or less.
Class Skills
The prophet’s class skills are Appraisal (Int), Bluff (Cha), Craft (Int), Diplomacy (Cha), Fly (Dex), Intimidate (Cha), Knowledge (all) (Int), Linguistics (Int), Perception (Wis), Profession (Wis), Sense Motive (Wis), Spellcraft (Int), and Use Magic Device (Cha).
Skill Ranks per Level: 4 + Int modifier.
Class Features
Weapon and Armor Proficiency
A prophet is proficient with all simple weapons, but not with any type of armor or shield.
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Spell Casting
A prophet casts psychic spells drawn from the psychic class’s spell list. To learn or cast a spell, a psychic must have a Charisma score equal to at least 10 + the spell level. The Difficulty Class for a saving throw against a prophet’s spell is equal to 10 + the spell’s level + the prophet’s Charisma modifier.
A prophet can cast only a certain number of spells of each spell level per day. Her base daily spell allotment is given on Table: Prophet (Same Spells per day as a Wizard). In addition, she receives bonus spells per day if she has a high Charisma score.
Prophets commune with their Familiar for their spells. Each prophet must choose a time at which she must spend 1 hour each day in quiet, trance-like communion with her familiar to regain her daily allotment of spells. Time spent resting has no effect on whether a prophet can prepare spells. A prophet may prepare and cast any spell on the psychic spell list, provided that she can cast spells of that level, but she must choose which spells to prepare during her daily communion with her familiar.
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Familiar
The Prophet must take a Familiar, and that Familiar must be chosen from one of the following: Life Crystal, an Aureole Crystal, or Calamity Crystal. While these are not living familiars, these items gain sentience and progress as a familiar of any other class.
If the Prophet chooses the Life Crystal as a familiar, the Prophet may add 1 spell from from the Cleric/Oracle Spell List to their Spells Known every time the Prophet gains a new level. Spells chosen in this way may be from the highest level of spell the Prophet is able to cast, but the Prophet has the option to choose Spells of lesser Spell Levels if desired. Special: If the prophet is evil, she may choose a Death Crystal that performs the same function as a Life Crystal.
If the Prophet chooses the Aureole Crystal as a familiar, the Prophet may add 1 spell from from the Sorcerer/Wizard Spell List to their Spells Known every time the Prophet gains a new level. Spells chosen in this way may be from the highest level of spell the Prophet is able to cast, but the Prophet has the option to choose Spells of lesser Spell Levels if desired.
If the Prophet chooses the Calamity Crystal as a familiar, the Prophet may add 1 spell from from the Druid Spell List to their Spells Known every time the Prophet gains a new level. Spells chosen in this way may be from the highest level of spell the Prophet is able to cast, but the Prophet has the option to choose Spells of lesser Spell Levels if desired.
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Detection Mastery (Sp)
At will, a prophet can use Detect Magic, Detect Good, Detect Law, Detect Chaos, Detect Evil, Detect Poison, and Detect Charm, as the spells.
At level 3, the prophet adds Detect Thoughts as an at-will spell. At level 5, the prophet adds Detect Desires as an at-will spell. At level 7, the prophet adds Detect Scrying as an at-will spell. At level 9, the prophet can perform any Detect spell she knows as if under the Still Spell and Silent Spell metamagic feats.
At level 5, similar to the Paladin ability, a prophet can, as a move action, concentrate on a single item or individual within 60 feet and determine if it is magic, good, lawful, chaotic, evil, and its thoughts or desires, learning the strength of its aura as if having studied it for 3 rounds. While focusing on one individual or object, the prophet does not detect in any other object or individual within range.
At level 11, she can concentrate on a single item or individual as an Immediate Action.
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Scrying Mastery (Sp)
Starting at level 1, as a 5 round cast time, the prophet may enter a quasi-trance with her familiar, and may focus on any creature or object the prophet names as if using the spell Scrying. At level 7, the Prophet always treats the object of her Scrying as if having Secondhand Knowledge, at a minimum. At level 14, the Prophet always treats the object of her Scrying as if having Firsthand Knowledge, at a minimum. At level 20, the Prophet always treats the object of her Scrying as if having a body part or lock of hair.
Each day she can use this ability a number of times equal to 1/2 her prophet level plus her Charisma modifier.
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Prophetic Knowledge (Ex)
Starting at level 1, and every level the prophet gains thereafter, the prophet receives two additional Class Skill points for free. These two Class Skill points must be spent on any Knowledge skill.
Furthermore, the Prophet may add her Charisma modifier to any single Knowledge Skill Check. The Prophet may use this ability 3 + her Charisma Modifier times per day.
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Twist Fate (Su):
Starting at level 1, at the beginning of your turn, you may, as a free action, roll a single d20. At any point before your next turn, you may use the result of this roll as the result of any d20 roll your allies or enemies are required to make as long as they are within 100ft + 10ft/level from you. If you do not use the d20 result before your next turn, it is lost.
You can use this ability a number of times per day equal to 1/2 your prophet level (minimum 1).
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At levels 1, 9, & 17, the prophet gains a bonus feat.
As far as a role within a group, the Prophet's job is kinda like the 3 Fates in the Disney cartoon Hercules. Every time Hades says something, the 3 Fates say "WE KNOW". The Prophet knows everything before anyone else.
As Chell Raighn said, Blood Arcanist can be a prepared full psychic caster. And yes, Arcanist is a prepared caster.
As for your other questions: Depends on what the author would want. PF has a semi-barrier of divine casters not being int-based and arcane casters not being wis-based (both are broken by two archetypes each, plus a bloodline for the latter), but psychic casters come in all ability score dependencies. A new class could use either the psychic list (to match Cleric/Oracle and Wizard/Sorcerer), or get a new list. As for what role they would fill, that, as always, depends on the player. Non-divine full casters are not intended as martials, but with a new spell list and/or class features, not even that has to be an insurmountable obstacle. Over-specialization on a specific role or playstyle is actually a very common mistake of homebrew classes.
Paizo doesn't seem to want prepared psychic casting at all, although I don't know why.
And by the way: [url=https://docs.google.com/spreadsheets/d/1ytw1h5A0Rl8JYmr9uV8MCcjKdk5YYr3CsV02T5zCGZg/edit#gid=0]Here is a spreadsheet of all spellcasters by flavor combinations.