Learn English – “Institution” and “facility”

I think both are wrong, if one equates use with acceptability.

In medicine, we say "he lost his faculty of speech", where faculty is the ability, power; an inherent capability, power, or function; a natural aptitude. Googling ___ gets ___ hits: "he lost the faculty of speech" (14.8K), "he lost his faculty of speech" (13K), "he lost the faculty to speak" (6), "he lost his faculty to speak" (1).

Facility is an ability to do or learn something well and easily; a natural aptitude. Googling: "he lost his facility to speak" (7 hits), "he lost the facility to speak" (0 hits), "he lost his facility to speak" (7 hits), and "I lost my facility for speaking* gets 3 hits, two of which are this question.

Psychological often refers to one's emotive and mental acuity state. Some people are psychologically better equipped than others when faced with hardship and particularly traumatic events in their lives, their minds will remain "sane" while others will suffer nervous breakdowns, fall into depression, or in more extreme cases, actually "lose their minds".

Mentally as an adverb, does not usually refer to one's sanity; to prepare oneself mentally before an arduous task is similar to psyching yourself up.

psyching yourself up: to try to make yourself feel confident and ready to do something difficult: I have to spend a little time on my own before I give a speech, psyching myself up.
C.D.O

mentally: connected with or related to the mind:
It's going to be a tough competition but I'm mentally prepared for it.
C.D.O

here the given example means to control one's nerves, anxiety, and emotional state. You mentally challenge yourself to face the task ahead. For some that task might even be traumatic e.g., speaking in front of a large audience, but it is a short-lived one, and people are rarely affected psychologically.

Psychically this one is trickier. First of all, as an adverb, it's not commonly heard in speech. Off the top of my head I can't think of an idiomatic expression which uses this term whereas its adjectival form, psychic, usually refers to a medium or any person "gifted" with paranormal abilities.

psychic adj : having a special mental ability, for example so that you are able to know what will happen in the future or know what people are thinking: psychic powers

psychic n: a person who has a special mental ability, for example being able to know what will happen in the future or what people are thinking: a gifted psychic
C.D.O

However, I believe, doctors and psychiatrists use the term psychic problems when referring to patients whose mental health is impaired or as many would say, are mentally ill. Why did I prefer the adjective mentally in that expression? I could easily have said psychologically ill and be understood, but it's a question of collocation. "Mentally ill" is by far the more popular term as the Google Ngram below confirms

Collocation

In Google books; "physically and mentally" yields a massive 10,200,000 results while "physically and psychologically" yields 1,340,000 results. If we add the term "prepare" the expression "physically and psychologically prepare" yields a modest 131 results whereas "physically and mentally prepare" produces a very respectable 1,510 results.

Summary

• How do you psychically prepare for pain before a surgical procedure?

Redditors were correct in saying that the term, psychologically prepared, used in that context is more idiomatic. Pain itself is not a task, a job, or a competition. To bear pain, especially prolonged pain, requires a healthy and rational mind.

• To perform a proper backflip, you have to prepare not only physically, but also psychologically

I wouldn't consider the term psychologically in this instance to be an error in the strictest sense of the word. It's fully understandable and I'm sure many native speakers have uttered similar things in their lives, nevertheless it's hard to argue against the term, mentally, being preferable. I would posit that performing backflips does not necessarily require a healthy, sane mind. It is a physical activity which requires a level of concentration and self-confidence.
^{(see: psych yourself up)}

Further data

Google books reports 9,960 results for "psychic problems"; 165,000 results for "mental problems" and 626,000 results for "psychological problems". By looking at their usage, the differences between these terms will be clearer.

EDIT

'Psychically' is better than 'mentally' in your first example, because it is that subconscious layer of the mind that one is addressing. Jon Jay Obermark

Whilst I may agree that the term, psychically, is not wholly inappropriate, psychically prepare is not idiomatic nor common. Indeed the Google Ngram chart shows that the expression psychically prepare is non-existent in American literature, whereas the expressions psychologically prepare and mentally prepare are both recorded.

The Chambers Dictionary 12th edition under psychic informs:

psy'chic adj. (also psy'chical) relating to the psyche, soul or mind; spiritual, spiritualistic; beyond, or apparently beyond, the physical; sensitive to, in touch with, or apparently having powers or capabilities derived from, something that has not yet been explained physically, eg apparently prescient or telepathic (psychical research investigation phenomena apparently implying a connection with another world; [...] ) psy'chically adv.