I wrote my letters myself too and used both. You need to show a successful track record and how it links to current and future work. But you also need to highlight the three prongs. Got some letters from previous professors who showed the success but also linked it to current work. Other letters mentioned the three prongs - worded every one differently so that it didn't look like I was working from script. Because I used laymens terms in letter of support, then when you come to write the cover letter you simply take these quotes and plug them in proving the three prongs.
Based on previous postings in this forum, the most important thing I learned was to ensure you make the case for third prong - the most difficult one. Show evidence how you exceed by far someone of minimum qualifications for this type of job. I showed that I was consistently winning contracts in a highly competetive field against other similarly qualified individuals, and that these contracts were being scrutinised by experts in the field. You must show evidence to back your claims up.
Hope this helps
Good luck