"Believe into Christ " and Believe into God" are exactly what the Greek says.
1Pe 1:21
Who by him
do believe into God (pisteuontas eis theon), that raised him up from the dead, and gave him glory; that your faith and hope might be in God.
John 3:15 That all
who believe into him (ho pisteuOn eis auton) should not perish, but have aeonous life.
16
He that believes into the Son (ho pisteuOn eis ton huion) has aeonous life.
Gal. 2:16 Knowing that a man is not justified by works of the law..... even
we have believed into Jesus (hEmais eis Christon IEsoun episteusamen) ..."
Many years ago I took the time to invest in a year of weekly night classes to study beginner Greek and Hebrew at a Bible college. I have been able to informally build on that basic understanding over the subsequent decades. I recommend anyone who wants to actually understand what the Bible is teaching should do likewise. If you are depending on translations to argue what scripture teaches, without being able to intelligently assess what the original language is saying, you are simply going to cherry-pick from translations to confirm your theological biases. Seek the truth, rather than merely to defend the uninvestigated opinions of your team.