Do I send my job offer salary counter email to only HR, or should I cc everyone in the original offer email: SVP and president/ceo?
If HR copied everyone to the original offer, you should keep everyone in the loop.
You could also call HR directly to discuss the counter-offer before answering the email. Call HR, negociate on the counter-offer, and then send an email as a follow-up.
EDIT : OP asked in a comment if he could avoid the phone part and stick with email.
It is OK to respond by email. However, if you have a counteroffer, I believe you and HR will need to discuss, and this will lead to HR calling you. You can avoid being the one to call, but if the counteroffer is significantly different, expect problems communicating solely via email.
Additionally, you mention being bad with the phone because you cave in too easily. You are coming in this discussion with a big advantage : you can prepare because you have full information. HR will probably have to deal with the surprise, and you will have your arguments prepared. Even if phone calls are not your preference, you have optimal circumstances to lead the phone call successfully.