I would say 'It depends' . You need to see the requirement. Both are for different purpose and ideally you or the concerned personnel must take a stand which is the priority - product quality or process quality .
Ideally I would take CSQA first and then CSTE , since process is advantageous than a product quality .
I began with the CSQA because at the time there was not a test for the CSTE. To obtain the cerification I would have needed to send documentation and work samples. I did not feel comfortable sending company data.
There is some overlap between the two certifications. The CSQA contains some testing and the CSTE contains some processes.
I would recommend starting in the field that you are working in at the moment and then sit for the second test the following year.
I have been manually testing web based applications for several years. I believe my testing ability is solid based upon developers feedback, but I lack process because my background was not in computer science and our QA department does not have much structure.
I do not think background matters when it comes to experience. It does not matter if you are a degree holder in computer science or not. What matters is how much experience you have gained while working. I may not ask an automobile engineer to open my car, but i will not question when a mechanic opens it.
If you have been doing testing for several years, you must be aware of the ins & outs of testing. For you CSTE would be best.
Second, since you will have to learn about processes while preparing for CSTE, you will gain that knowlede as well. This may prove as an opportunity for you to introduce some structured processes in the company.