What is the ruling on using "bi'idhnillah" (by Allah's permission) instead of "insha'allah" (if Allah wills) when intending to do something in the future? And does the former link the action to Allah's will as the latter does? And is there any presumption against Allah in it?

The default for a Muslim, when referring to future events, is to say "In sha' Allah" (God willing), following the example of the Quran and Sunnah. There is no objection to replacing it with "bi idhnillah" (by God's permission), as it conveys the same meaning, though adhering to the ورد (what is stated in the texts) is preferable. A servant does nothing and performs no act of obedience except by God's permission. The concepts of "mashi'ah" (will) and "idhn" (permission) are close in meaning, as both are cosmic and legislative. The phrase "bi idhnillah" is equivalent to "in sha' Allah" when the intention is to attribute the matter to God's will.