The key to this issue is to make a distinction between what to call the process of relying on emotional preference rather than evidence as a basis for a belief, versus what to call the belief itself that results from such a process.
Wishful thinking has the proper connotation of referring to a process of cognition in which one's wishes/emotional preferences for what a person wants to be true were used as the basis for concluding that something was actually true.
The reason that the word "faith" gets messy is that it is used to mean both of these, and sometimes used simply to mean that one has a strong belief in something, without implying anything about what that confidence in based on.
"Relying on faith" as a method of deciding what is true (which religions promote as an epistemology) is a reference to the process of relying basically on emotion or deference to authority decrees rather than on reasoned analysis of the evidence. That notion of faith is essentially wishful thinking.
That is why the founder of the Protestant Reformation, Martin Luther, said that "Reason is the greatest enemy that faith has."
But a phrase like "I have faith in you my son." just means to have confidence in a person, and that confidence might be rationally based, such as in the fact that the person has always come through in the past.
Delusion usually refers to a belief itself, implying that belief was not only not the product of reason, but that their is clear evidence right in front of the person that their belief is false, yet they remain firmly committed. I would argue that this also true of most religious beliefs, which is why religion developed the whole notion that relying on faith is a virtue, because one must commit to using wishful thinking in order to maintain belief in such absurdities that are so clearly false based on basic reasoning. The association with clinical mental illness seems inaccurate. The mentally ill will have delusions that are uncommon and not shared by many others and with no obvious emotional benefit or motivation for maintaining the delusion. Their delusions are not fueled by some self-serving wishful thinking, but arise from uncontrollable factors that undermine their ability to reason. With wishful thinking, one can create delusions, but they are almost willful delusions where the believer could choose to apply reason and the belief will go away, but they deliberately repress the application of reason to protect cherished beliefs.
Intuition is something else. It basically means that the idea did not arise from a process of fully conscious formal rational analysis. Intuition means that a idea vaguely seems correct to the person, but the source of that sense of correctness can vary and is not inherently emotional. An idea can relate to some knowledge you have but you haven't formally analyzed whether all your relevant knowledge actually supports it. For example, you might meet a person and based on their body language and the first few things they say, you form an intuition about the type of person they are and the type of action you would expect from them. OTOH, since its a vague sense, it can be biased by emotional preferences too. That's why wise people do not blindly trust intuition, and attempt to verify it when they can. Which speaks to the fact that intuitions don't really qualify as beliefs because often the person who has them has little confidence in them. Subconscious processing produces intuitions all the time in all people, but people vary in whether they just go with the intuition or evaluate it more deliberately.
