In the absence of a wiki, here's my attempt to a step by step guide for a topic that concerns so many people, but it's needlessly overwhelming when you are in an acute crisis.

This is, to my knowledge, the fastest way to get insurance-approved psychotherapy in Germany.

1. Book your first talk session over the 116117 Appointment Service

• On the first time there, when asked Haben Sie einen Vermittlungscode? [Do you have a referral code?] select Nein [No]
• Next, when asked Wählen Sie die gewünschte Fachgruppe und ihre Postleitzahl aus [Select the desired department and its postal code] select Psychoth. Sprechstunde Erwachsene [Psychoth. Consultation hours for adults] or Psychoth. Sprechstunde Kinder & Jugendliche (bis 21 Jahre) [Psychoth. Consultation hours for children and young people (up to 21 years)] depending on what applies to you. For PLZ enter your local postcode. This is for sorting and limiting the results by location.
• In the next two fields enter and re-enter your email address and tick Ich bin gesetzlich krankenversichert. [I have statutory health insurance] if you are insured by one of Germany's public health insurance funds (such as TK, an AOK, a BKK etc).
• You will shortly receive an email titled Vermittlungscode für Ihre Terminbuchung [Referral code for your appointment booking] and a link back to the 116117 Appointment Service
• Visit the Appointment service again, but this time choose Ja [Yes] to the question about a referral code.
• Copy your referral code and enter your postcode and hit Termin suchen [Search for appointment]
• If asked to select the department again, select the same option as before, e.g. Psychoth. Sprechstunde Erwachsene
• Results should show up now, see step 2

2. Selecting your first therapist; no need to think about it too hard at this stage

The results you will see this first time are psychotherapists who can offer you this first talk session and a provision diagnosis/recommendations for further care. This availability does not imply that they can take you as a long-term patient so there's no reason to think too hard about who to choose.

• While you are in the results page, two icons at the top allow you to adjust the search radius (default +5km from the centre of your postcode) and availability (defaults to both mornings and afternoons all seven days of the week)

• In the list below, you can select upcoming appointments, separated by date and sorted by distance. Those can be on the same day or a couple of days in the future. You may select one appointment.

• It bears repeating that this choice shouldn't paralyse you. The truth is, it's very likely that you will only see this therapist once. The most important selection criterion at this stage is that you share a language that you both speak very well. The Appointment Service doesn't offer that information, so you will have to use an external website to check the resumes of the therapists you see.

A comprehensive (but not 100% complete) catalogue can be found at therapie.de. I suggest you look up all names you see in the Appointment Service there. One way to do it is to go to your preferred search engine and search for site:therapie.de FULLNAME.

If they have a profile there, under Stichpunkte [Key Points], look for Weitere Sprachen [Additional languages].

There's more information in therapie.de but it will only be really important later.

• If you have found a therapist that is available in the next days and you share a language, book the appointment for your next convenience. Don't hesitate to call sick at work if necessary, but if you need a doctor's note, be aware that therapists cannot write such notes, you should reach to your GP or other specialist that is aware of your mental health concerns.

• The appointment service asks you to directly reach out to the therapist and confirm your booked appointment. Some therapists have specific preferences, but most actually prefer email or leaving an answering machine message because they do not have secretaries so they can't answer calls most of the time.

3. At the first talk session

Congrats, you completed the most difficult part: starting this whole process.

• What you absolutely need to have with you is the health insurance card. Any additional documentation you have can help, but it's not necessary.
• At your first session, arrive 10 minutes early and wait to be allowed in. Most therapists work out of residential buildings and do not have secretaries, so if this is your first time visiting their practice, it might feel a bit confusing and unusual.
• After taking your health insurance card and asking you to fill the patient information sheet, the therapist will ask you what brings you there. Feel free to ask them how they want the information to be ordered. Personally, I follow the reverse chronological structure (a) immediate/most recent issue/acute complain, (b) life experiences in the last few years including all mental health episodes/crisis points, (c) general life situation, (d) goals you want to achieve if you can start therapy now
• At some point, the therapist will start asking you questions in order to establish one or more Verdachtsdiagnosen [provisional diagnoses]. If this is your first time in therapy, keep in mind that the therapist is not questioning you because they are suspicious of you, but because they need to tease apart similar diagnoses and also the severity of each one. If you don't understand their questions, ask for clarifications. It's especially important if both of you are talking in a language that is not either one's native one (e.g. if you both speak English as a second language).
• Near the end of those 50 minutes, the therapist will fill the very important form PTV 11 for you - this form contains your provisional diagnoses, the recommendations for further treatment (including which type(s) of psychotherapy can work for you), the level of urgency, the potential need for the involvement of other specialties such us your GP or a ~~psychotherapist~~ psychiatrist (if medication can be potentially helpful, for example), and a new referral code in case they cannot take you on as their patient. Ask the therapist to explain to you all items on that form. They will gladly do so in most cases, and even if they don't, remember that you only asked them to do their job, so you are not wrong for asking.
• It's very likely that they cannot take you as their patient at this point. If you are lucky and they have availability, you can consider having a small number of probationary sessions with them to see if you are a good match. It's extremely important that you are a match, so if it's not working out, do not hesitate to ask for a referral code in order to seek a new therapist. There's no reason to feel bad about doing that.
• If they tell you that they cannot take you on as a patient though, ask them to recommend you some of their colleagues who they think may be good matches based on your profile and diagnoses. Such targeted recommendations are more efficient and sometimes they can even direct you to colleagues who they know have availability or they are going to have availability in the next few weeks.

4. Booking the next session

Assuming that you need to find a new therapist after getting your provision diagnosis, you can start by directly calling or emailing the recommendations you got (if you did), and also use the new referral code on the appointment service. Therapie.de is going to be useful to you again, as it can also act like the yellow pages for your search.

This part of the process can be extremely short or extremely long, and it's important to keep it up. Here, you may consider an accountability structure, such as having a friend with you when you are calling/emailing, who can motivate you to keep going.

5. At the next few sessions

You now have a therapist with availability or who is going to have availability soon so they can offer you some probationary sessions until the availability is there.

Different psychotherapy approaches apply here and it's out of scope to discuss them, but before the fourth session you will need to carry out one last bit of paperwork for the health insurance.

• Your therapist will give you the form 22 Konsiliarbericht [consultation report discussing the indications for psychotherapy] which your GP has to fill and which an independent expert will evaluate before telling the health insurance a simple Yes or No to covering your therapy.
• With a medium degree of urgency, arrange to see your GP (or a new GP you locate now) and take this form with you. You will discuss all relevant points with your doctor and they will fill the form for you to return to the therapist.
• It's possible that the GP will order some lab tests, especially a thyroid functioning test. Abnormal thyroid functioning can cause mood disorders so at this stage the health care system wants to catch any possible physical causes before moving to the mental ones.
• Once completed, return the form to the therapist. They will handle further steps.
• Your therapist will let you know for how long and how often you will see each other. The length of treatment can be extended base on need, and a usual rhythm is meeting once weekly.
• It's harder to arrange all your appointments to be outside of work hours, so if your therapist can offer you a stable day and time week-in week-out, it's a good idea to talk with your employer about a permanent change in your work hours (if necessary).

I hope this can help anyone who stumbles upon it. Please mind that I can't help you further than that, because the above is my entire knowledge around the process. Feel free to point out any mistakes too.