I'm a professor at a public university and I coach our upcoming grads on contract negotiations each year.
Your approach of telling what you would need to move to the position is great, especially because you laid out concrete reasons for the higher salary.
I'm not sure if you've been communicating with the recruiter or who will ultimately be your "boss", but depending on the size of the company, the recruiter may have very little control over what package they can initially offer you. Recruiters and even HR are usually strictly limited to initial target offer. The concessions to you during the negotiations will usually have to come from your boss (your direct report, division head, etc) or the business manager of the division (if the company is large enough).
Being a strong advocate for yourself and being upfront is a great strategy and works for many people. So many of my young grads are afraid or uncomfortable with laying it out there what they want, and end up selling themselves short.
Job negotiations can be emotional for the employee but you must remember it is just business, especially for the employer. As long as you negotiate in good faith, it is very rare an applicant to blow it or insult the hiring company. The key is negotiating in good faith though. One warning I would give is that you mentioned that you weren't even looking for the new job, they approached you. However, if the company came back to you with the terms and salary that you asked for, and you still turned them down, that may be the one situation that may insult someone. I tell my grads that it is perfectly ok to interview at many places and field many offers, but you ask yourself once you make a counteroffer and the company gives you everything you asked for, are you ready to accept the offer?