How is it proven when implantation occurs? different things are said: 3-5 days after ovulation, 6-10 days...?

Very good question: Historically, based on 1) studies of 'endometrial recptivity', 2) non-human primates (rhesus monkeys), and 3) embryo transfer in ivf. It's further complicated by dividing 'implantation' into phases. Currently the first phase is called adplantation and occurs after 6-7 days. The work of bergh and navot, 1992 is the foundation of endometrial receptivity http://humupd.Oxfordjournals.Org/content/13/.