Salaam Shakir, you had asked for the difference between Billahi and Bismillahi:
Research source: Ma’ani Al-Huruf by Rommani
Remark: This is a short memorandum to address the difference but a more detailed treatise could be written if there is interest. The research is quite involved and requires much effort to present.
A. Ba sound in Billahi indicates a Qasam (Oath)
B. Ba sound Bismillahi is a declaration of commencement a form of Dhikr (Remembrance)
The two differ significantly.
Qasam (Oath) is uttered when the speaker needs to render the veracity of a statement made by him i.e. immense personal belief in that statement (‘Aqida firm belief), and do so by mention of another statement which is held dear/noble/true by the listener or by himself e.g. On my children’s soul!
Therefore Billahi is a Qasam (Oath) you use it when you want to tell me convince me something you believe 100% or something you will do or did for real, and you mention Allah’s name after the Ba sound, then I know you are telling the truth.
On the other hand, Bismi is a standing phrase the Arab used to commence an offering e.g. offer something to a noble or idol by using Ba sound to commence the offering juxtaposing the name of the noble or idol after Ba sound, so he would say Bismi so and so I offer this sacrifice… commencing slaughtering the animal or bring some gift closer and so on.
Since the advent of the Prophethood of Muhammad peace be upon him, the term is almost exclusively used for Allah’s names in particular Rahman and Rahim, thus no idols or nobles names allowed.