Sorry, yes I was up early with the puppy! Will try and do a potted summary of the main ones. You forget when immersed in it all.
CBT = cognitive behavioural therapy based around working towards behavioural and attitudinal change (type of therapy common in the NHS)
EMDR = eye movement desensitisation and reprocessing. Therapy offered to help with PTSD and complex trauma among other things.
Systemic therapy is a type of family therapy that might be offered to couples, family groups or even workplace teams. Can be done individually as well. Very crude explanation of it is the basic premise that we can only understand self in relation to others/the systems we operate within. A family could be a system, society is a system, a church/religion, a workplace or school, a friend group, etc.
Core training = the training someone does to qualify them to practise independently as a counsellor, psychotherapist, psychologist, etc.
Post qualifying specialism = a different therapy approach or specialism (may also be called modality) that you can only do after completing your core training.
CYP = children and young people
CAMHS - the NHS Child and Adolescent Mental Health Service. Though "lack of service" may better describe it these days!