Curious as to why you do not want to register online (assuming that you are registering at all). In my experience, anything that cuts down on the human factor at HMRC is a bonus.
BTW the quasi-employer may prefer to treat you as self employed for various selfish reasons: He avoids secondary class 1 NIC, he avoids a bit of admin, and he avoids a huge raft of employment legislation which provide you, the employee, with significant rights (rights to tenure of employment or redundancy pay, sick pay, paid holidays, working hour commitments, national minimum wage restrictions and so on). It would well be the last of the three that is the driving influence.
Whether you are in fact an employee is perhaps not a matter of choice, although the employer is taking most of the risk. But not all. As a self employed individual you may have more generous rules for what is deductible from your income. While the PAYE/NI would be down to the employer, you could still have some of your deductions disallowed retrospectively with penalties, and in law the employer may have some redress against you for arrears of PAYE on which they are assessed (collecting it would be another matter). It all gets very messy, because you will have paid some tax on the self employed earnings that are being recategorised as employed earnings.
As to whether you are in fact employed, the fact that you are only working for one customer is influential but not conclusive, as the questions in the ESI tool demonstrate.
BTW there have been a number of court cases, not tax-driven, in which individuals who have throughout the engagement been treated by both parties as self-employed, but where the "employee" has sued for employment rights, typically redundancy on dismissal, on the grounds that whatever the agreed treatment for tax the reality was one of employment. Been quite successful, as it turns out.