Re: Which calender system is Islamic?
هُوَ الَّذِي جَعَلَ الشَّمْسَ ضِيَاء وَالْقَمَرَ نُورًا وَقَدَّرَهُ مَنَازِلَ لِتَعْلَمُواْ عَدَدَ السِّنِينَ وَالْحِسَابَ مَا خَلَقَ اللّهُ ذَلِكَ إِلاَّ بِالْحَقِّ يُفَصِّلُ الآيَاتِ لِقَوْمٍ يَعْلَمُونَ
10.5] He it is Who made the sun a shining brightness and the moon a light, and ordained for it mansions that you might know the computation of years and the reckoning. Allah did not create it but with truth; He makes the signs manifest for a people.
Both calendars can be used, in my personal opinion Solar is good for calculating a day and Lunar for month.
In Lunar calendar just by viewing cresent we can mark first day of month and similary from waxing to full moon and then from waning to its disappearing we can calculate complete month.
Even though you've mentioned moon was created as 'Badr' but it doesn't mean we should use 'Badr' as first day of month, like Adam was created as an adult but life span of a normal human begins from birth. You know what I mean.