The true definition of Bid’a is ‘to initiate or create something upon an unprecedented example’. In the Quran, Allah describes himself as the ‘Badi us-Samawaat wa al-Ardh’, meaning the creator if the skies and earth. Badi’ here means a creator who brings something into existence from no previous material. On this basis, Bid’a, which comes from the same root word as Badi’, means to initiate or create something upon an unprecedented example.
As you have pointed out, not everything that is Bid’a in Islam is prohibited. The Prophet (peace and blessings of Allah be upon him) gave the best definition and criteria for deciding what innovation is commendable and what isn’t. He said,
“Whoever innovates [something] which is not from our matter [of Islam] then it is rejected” (Bukhari and Muslim)
What this saying clearly indicates is that not all Bid’a is rejected; rather those innovated practices are which are alien to Islam.
Furthermore, the Prophet –in another saying- outlined the fact that there are some innovations that are actually good, and will lead to a reward. Imam Muslim narrates a Hadith through the chain of Jarir ibn ‘Abd Allah, who reports that the Prophet (peace and blessings of Allah be upon him) said,
“Whoever initiates a good practice in Islam then he will attain the reward for it and the reward of whoever acts upon it thereafter, without any shortcomings. And whoever initiates an evil practice, then he will carry the burden as well as the burden of whoever acts upon it thereafter, without any shortcomings”
What this Hadith thus shows is that not all innovation is contrary to Islam. If this was the case, then the Prophet (peace and blessings of Allah be upon him) would not have promised a reward for those who initiate a good practice.