Let us find out the origin of difference between bhaava-madhya and bhaavaarambha. There was no difference between Ptolemaic and Vedic horoscope because ayanamsha was almost zero then. There are irrefutable evidences from Arabic and European sources that no astrologer or astronomer used tropical zodiac before 15th century. The value of ayanamsha in Handy Tables of Theon of Alexandria, of Baghdad's astronomers, as well as in Alphonso Tables of Spain and other tables of Europe including that used by Copernicus was always almost equal to Suryasiddhaantic ayanamsha at their times. I possess irrefutable evidendes of these facts. Unfortunately, opponents of Vedic Astrology are spreading lies during modern era and propagate that tropical system is eternal. But more serious writers in West accept that the most ancient sources of Western Astrology were always sidereal : Mesopotamian (Chaldean) clay tablets in cuneiform script which have been translated and published (I possess its digital copy, sent to me by a Turk from Intanbul).
It was around 15th to 16th century that traditional astronomy was rejected in favour of Heliocentric tropical system. Gradually, astrologers, starting from Kepler. started shifting to tropical system. I do not know which system Kepler used in astrology (he was an astrologer primarily, which modern historians hide because they hate astrology), but he was the first person who sowed the seeds of tropical system in astronomy, strengthened by Newton firmly much later.
Till that time, the value of ayanamsha was about 15 degrees (1499 AD to be exact). Hence, when sidereal system was discarded in Renaissance, it was necessary that 15 degree offset ought to be added to ascendant for making correct horoscopes. Hence, bhaava-madha became bhaava-arambha. For instance, if Lagna was at 75 degrees in tropical system, it should be 60 degrees in sidereal system at that time due to 15 degree ayanamsha. Therefore, if sidereal horoscopes are changed into tropical, sidereal lagna of 60 degrees at bhaava-madhya will be same as 60 degree tropical lagna at bhaava-arambha.
Unfortunately, later Wetern astrologers stuck to this mediaeval. Hence, with nearly 23 or 24 degree ayanamsha now, there is 7 or 8 degrees error in western horoscopes, which resuslts in 25% wrong horoscopes if purely tropical system is used with lagna as bhaabaarambha (if tropical system is used with lagna as bhaava-madhya error will increase to 80% or 24 degrees out of 30 deg in a sign). 25% error is overlooked, but it makes a nonsense of divisional charts in which errors multiply. Hence, vargas cannot be used in western system. Instead, they invented harmonics which has no basis in classics of east or west.
With increasing ayanamsha, errors in tropical system will continue to increase and a time will come when all of them will revert to sidereal system. That time is not too far off, because after 500 years error will go up to 50% which will make a nonsense of tropical astrology.
Till ~1500 AD, sidereal system was in vogue in Europe as old records show. And bhaava-madhya of first house was lagna as in Vedic system. But when Heliocentricism triumphed in astronomy, tropical system was adopted, as in calendar in 1582 AD. Suppose for some moment lagna was at 60 degrees sidereal (bhaava-madhya), with 15 deg ayanamsha tropical bhaava madhya would be 75 degrees. Then, tropical bhaavarambha would be at 60 deg.
Hence, there would be twp ways of shifting from sidereal to tropical. One way is to add ayanamsha to sidereal and take bhaavamadhya as lagna. Another way was to take bhaavarambha as lagna without changing the magnitude of sidereal lagna degrees. Sidereal 60 deg bhaavamadhya would become 60 deg tropical bhaavarambha.
I have not been able to find direct evidence of such a shift. But before 1500 AD, everywhere in the West ONLY sidereal system was in use and I have failed to find any trace of tropicalism, in spite of false claims of modern tropicalist astrologers. Even Copernicus used sidereal system with almost Suryasiddhantic ayanamsha. And tropicalism triumphed in 16th century, culminating in Gregorian Reforms in 1582 AD. That is why the shift must have occurred around 1500 AD. Then, ayanamsha was at 15 degrees. At about the same time, shift from bhaavamadhya to bhaavarambha took place in the West.
They did not place the cusp at the beginning. Actually there was nothing like tropical astrology anywhere in the world. Tropicalism evolved in 16th century due to pressure from heliocentricists who wanted everything in the sky to be measured with reference to the ecliptic, which is an absurd idea because Sun is not the centre of the Universe, while Fixed Sky of siderealists is a better frame of reference.
I am re-writing your expression for early 16th century practice :
"We would say that bhav madhya is mesha 2. Our bhav will extend from from sidereal Meena 17 to Mesha 17.
For the tropicalists, since they place the cusp at beginning of a house, the bhav will extend from tropical Meena 17 to Mesha 17".
With 15 degree of ayanamsha, both systems would give exactly same positions of all planets and houses, and astrological fruits will be the same. The only difference is that what we call house-middle in sidereal system is called house-start by tropicalists. That is why the fundamentals of Vedic and Western astrology are the same, such as topics of twelve houses. It is universally acknowledged by all western scholars that both system have same origin, and it is also well known that the origin was sidereal, traced to Chaldean astrology by westerners (because they do not recognize Indian records as authentic due to cultural prejudice against a poor and defeated nation). But some fools are falsely propagating that the present tropical astrology is the original system, although there is no evidence of such an idea.
Therefore, when tropical astrology arose in early 16th century, it was astrologically correct as far as houses were concerned. But two significant errors cropped in which were not solved or even noticed because of two factors : Church disparaged astrology and banned it in most of western nations as a result of which , and merit flocked to modern science, hence only fools flocked to astrology in the West. Therefore, places of exaltations of planets or their own signs , which were almost identical in Ptolemy's system and BPHS became distanced by the amount of Ayanamsha. For instance, both Ptolemy and BPHS say that the ecxalted point of Sun is at 19 degrees of Mesha. But now, tropicalists put it at 10 deg tropical Mesha. Second error is due to continuous increase in ayanamsha. When ayanamsha was 15 deg, bhaavas were correct in tropical system and had no astrological difference with sidereal system, but with 22.6 deg ayanamsha (siddhaantic) there is 7.6 deg error in tropical houses and therefore 7.6 out of 30 degrees is the amount of error in houses. Hence, 76 out of 300 charts or 25% are wrong in tropical system. This error continues to increase, till it will approach about 50% when tropical system will become useless for D1. For higher divisionals, it is still useless.