You have quoted an interpretation from a psychologists personal website business.
"Paul is a Psychologist who specialises in the treatment of Depressive and Anxiety disorders."
I would cite that the wikipedia's statement of CBT is more accurate and less personalised to this persons specific business model, being:
Cognitive behavioral therapy (CBT) is a psychotherapeutic approach: a talking therapy. CBT aims to solve problems concerning dysfunctional emotions, behaviors and cognitions through a goal-oriented, systematic procedure in the present.
EMDR is a CBT therapy, which you cite against CBT, and it achieves the exact same thing in a different manner.
CBT is a model which works like most therapies, being present tense, because empirical data has found that little is achieved by taking a person in the past.
However, therapy models also typically dictate that in some cases a person must still be guided back into the past in order to process a problem, but you don't let them dwell there, instead bringing them back to the present.
You raised TIR as an example of inside out, yet from the TIR association they state the same principle as for what the majority of trauma therapies cite:
When something happens that is physically or emotionally painful, one has the option of either confronting it fully and feeling the pain, or trying in some way to block one's awareness of it.
Confronting any type of trauma is called exposure, and TIR replicates more PE than any other therapy model, which is one part of a complete CBT approach.
It would actually be difficult nowadays to honestly tell what model a therapist is using, because there are literally so many rolled into one, what do you really call it? Bitza Therapy? Trauma therapy really is bits of this, bits of that. The moment you engage a client using reflective listening, you're using Person Centered and its gaggle of names used depending on location in the world, and not CBT. If you coach them in a life skill, then technically that falls into NLP, being business oriented coaching / communication skills.
I think its hard to really identify a specific name nowadays, and I think its more why CBT is merely used as a term more than really the adage 12 session therapy model it theoretically is.
Give TIR to 5 different therapists and I guarantee you will get 5 different interpretations of how they want to deliver that therapy model. One or two may get it right, all five may get it right, all five may get it wrong. No different to any therapy model. Do some reading here on people's bad experiences with dodgy EMDR practitioners. The model is sound and proven. Put it into a therapists hands who has decided to add their own personality to it, and suddenly they're screwing people up worse than when they came in the door.
Moral of the story. There is no sound, one solution fits all approach in treating trauma. DBT is a CBT therapy, yet touches little on the trauma and focuses heavily on emotion regulation and dissociation grounding techniques for BPD.
The facts though, is that there are four proven effective therapies for the treatment of PTSD, which are: Cognitive Trauma Therapy, PE, Stress Inoculation and EMDR. These four have proven without a doubt the best treatment options for PTSD symptom reduction longevity. No quick fixes... they all target the long term, not the short term.
Short term therapies (quick fixes) are not a viable solution for PTSD.