[RPG] What non-explicitly-psionic Prestige Classess advance psionic manifesting

Yes, they stack like the rule you cite says, if the specific class says they do. This is an example of exception-based design - the general rule is "only the one class counts" but then the specific rules in the other classes can say "but I do."

Not all prestige classes are alike so you should carefully read their Spellcasting benefit. In this case, Arcane Trickster says "I stack."

But classes that do not say that phrase, like Wizard and Sorcerer, do not stack. In your second example, Loremaster's spellcasting is worded like Arcane Trickster (as if she had gained a level in a spellcasting class she belonged to...), and you would need to decide whether you want it to stack onto your 5th level wizard spellcasting or your completely separate 3rd level sorcerer spellcasting. Assuming the former, you'd cast your spells from wizard slots at level 8 and your ones from your sorcerer slots at level 3.

It's kinda obvious why this is, the core idea behind prestige class design is "continue what you were doing, down a slightly weirder path" and therefore completely penalizing a spellcaster by e.g. halving their spellcaster abilities would be undesirable (even at that, many don't add a spellcasting level at every level and have one or two "penalty" levels.

A bloodline level grants no increase in base attack bonus or base save bonuses, no hit points or skill points, and no class features. It counts as a normal class level (with no class skills) for the purpose of determining maximum skill ranks. Levels of bloodline never result in XP penalties for multiclass characters.

Include the character’s bloodline level when calculating any character ability based on his class levels (such as caster level for spellcasting characters, or save DCs for characters with special abilities whose DCs are based on class level). The character doesn’t gain any abilities, spells known, or spells per day from the addition of his bloodline levels, though—only the calculations of his level-based abilities are affected.

If a character has levels in two or more classes in addition to his bloodline levels, each class gains the benefit of adding the bloodline levels when calculating abilities. (19)

The extra levels/spells the prestige classes provide are class features, so would not proc with the levels of bloodline. As far as In know, only the primary class counts towards the caster level unless the class features of the prestige class explicitly add to it.

Unless I'm missing something...

While bloodline levels grant no class features, the bloodline levels do still exist. This answer would be more useful if added to it were a more detailed explanation as to why a character should not "[i]nclude the character’s bloodline level when calculating... caster level" if the prestige class grants caster levels. (Note that were they to have any impact, the bloodline levels would only improve effective caster (or initiator or manifester or soulbindermanifester or whatever) level, not, for example, spells known or spells per day.) – Hey I Can Chan

Let's say Wizard is your primary spellcasting class. Prestige levels have no effect on your caster level. Some prestige classes have class features that improve your Wizard's caster level and adds some spells, etc. However, since bloodline levels do not grant these features, they do not effect the Wizard's caster level. The bloodline levels would increase the caster level for the Wizard though as the caster level is calculated by the number of levels the character has in Wizard. In any prestige classes in the official rules I can find, it explicitly states a prestige class adds CL. Take Eldritch Knight as an example:

Spells per day: From 2nd level on, when a new eldritch knight level is gained; the character gains new spells per day as if she had also gained a level in whatever arcane spellcasting class she belonged to before she added the prestige class. She does not, however, gain any other benefit a character of that class would have gained (bonus metamagic or item creation feats, bard or assassin abilities, and so on). This essentially means that she adds the level of eldritch knight to the level of whatever other arcane spellcasting class the character has, then determines spells per day and caster level accordingly.

(emphasis on the last line mine) The above is a class feature, so would not be affected by bloodlines. Meaning that a level 2 Eldritch knight would only add 1 level of spells and caster levels as the class only has 1 stack of this class feature at level 2 and doesn't gain another stack through bloodlines.