Some say that Sachi approached Vishnu to be her lover, but He refused but made her take birth as Radha in her next life. Is it true or is it some fake story created by the British?
The story seems to be from the Panduranga Mahatmya of the Padma Purana, which tells the story of the Krishna deity of the Panduranga Vitthoba temple. Here's how page 22 of this book summarizes the Panduranga Mahatmya's account:
It begins in Vaikunta. The time of action was a pleasant dusk. Sri Mahavishnu was seen absorbed then in the colourful spectacle of the west, bathed in the mellowed light of the setting Sun, caressed by cool fragrant breeze, coming over His Nandanodyan, when Sachidevi, wife of Devendra stood before him. All alone in her full splendour. Up from his dream world, he requested with all decorum due, to inform him the reason of her surprising visit, unattended by any companion. And without appointment at it. Quivering all through, as shyness enveloped her whole frame, she let out that she had been pining for amorous union with him, though she knew, it was unbecoming on her part to entertain such lewd desire and it was rather sinful too. Even though she tried all means to suppress that impious instinct, the temptation of his jaganmohanakara was irresistibly too much for her, and so there she was. But Vishnu politely and sincerely rejected her advances, as it would prove disgraceful to both. Moved by the moist eyes, he said that her desire would be fulfilled in her next birth, when he would be born as Krishna and she as Radha. Despaired Sachi devi left him at once.
Page 907 of this book claims to give an exact quote of the Panduranga Mahatmya's account:
Once the wife of Indra, the king of gods, named Sachi, approached god Vishnu. She wanted him to fulfil her lustful desire but failed. God Vishnu, however, promised her that he would fulfil her desire in the next incarnation. Accordingly Vishnu became Krishna and Sachi became Radha. Both of them were living in Mathura. Rukhmini, the wife of lord Krishna, one day caught them sitting together and got furious. To teach a lesson to her husband she left Dwarka without anybody's knowledge. Her husband, on noticing this, started for her search. During the course of his search he came to a place called Dindeervan where he found her engaged in meditation. He also started penance. Both of them would have continued with the tapa (penance) but for the intervention of Pundlik.
But I can't seem to find the Panduranga Mahatmya in the Padma Puranas, so it's possible that it's an interpolated section. And this whole story rests on a premise that Radha and Krishna reunited at some point in Mathura/Dwaraka, and as far as I know there's no scriptural basis for this encounter. It is mentioned in folklore though, as discussed in this book.
In any case, Radha is usually not thought of an incarnation of Indra's wife Shachi, instead she's either thought of as an incarnation of Vishnu's wife Lakshmi or Vishnu's wife Niladevi, or she's conceived of as the eternal consort of Krishna in Goloka by Gaudiya Vaishnavas and others.
No at the time of Krishna , Indrani Sachi wasn't reborn as Radha. But as told in Adi Parva /Sambhav Parva -Chapter 67 - Shloka -157 - ( See P145 ) Of Mahabhrata ,Draupadi was Amsha Avatara of Sachi.
द्रौपदी त्वथ संजज्ञे सचीभागदानिन्दिता |
द्रुपदस्य कुले कन्या वेदिमध्यानिन्दिता ||157||
And the faultless Draupadi, slender-waisted like the wasp, was born of a portion of Sachi (the queen of the celestials), in the line of Drupada. Through medium of the Yajna-Vedi.
So the story of Vishnu made Goddess Sachi to take birth as Radha during Krishna's life doesn't seems to be much authentic . Below is screen shot of hindi translaton.
According to Prof. M A Venkatakrishnan and other scholars, Radha is not mentioned anywhere in Srimad Bhagavatham. She has been popularized in Jayadeva Ashtapadi. So, there is no way she can be a reincarnation of anyone.
Prof. M. S. Rangachari of Triplicane told me that the root of the name Radha means sorrow or concern, symbolizing the Supreme Being's concern for the souls trapped in samsaram.