Note: Many theologians, priests, rabbis, imams, and believers in God can argue that your situation mirrors life right now. I feel the Worldbuilding Stack Exchange is not the place for this religious discussion. This answer will attempt to answer your question and nothing more; rephrasing this question and posting on Mi Yodeya, Christianity, or Islam SE may yield more insight into this.
It seems you have several attributes for your deity that you want to enforce. Among which:
- Freedom to choose and of belief is granted to intelligent beings
- Allow for reasonable deniability of the creator's existence
(I should mention that the groups mentioned in the starting note may say things about why it is this way, but that's not here nor there for the scope of this answer.)
This is actually easy; this deity ought to give believers miracles which is in line with their faith and that god's will. However:
- These "miracles" need not be supernatural occurrences, but unexpected or direly needed by your believers.
- To retain reasonable deniability, not everything should go their way. That is, the more subject believers are to forces and events that occur to everyone, the better.
The key here is that the miracles in question can be explained by "scientific" or "rational" thought. If a person can just as easily look at isolated incidents and say "that was chance" or "it just happened to be that way" as say "that was my god," then people can argue for and against the existence of that god. This can result in the same problem posed by the Bielefeld Conspiracy, except it's a deity and not a town in Germany; how do you prove that something exists without witnessing it?
What About Science?
If this god wants to avoid being exposed by scientific inquiry, you simply need to avoid repeatable responses and bring your miracles about by mundane, natural-law abiding means. My wife is a grade school science teacher, and the public-school approved curriculum teaches that things which cannot be repeated are not fit for scientific inquiry. (As another side note; lack of repeatability may be why psychologists have such trouble replicating results.)
Reliance On Deniability
Of course, you shouldn't ever discount people's ability to deny. Some people just don't seem to accept some things even though it stares them in the face. People can by annoyingly or unreasonably skeptical, especially when suffering an existential crisis. After all, if you're unsure if you exist, how can you determine if something else does?
Also, remember the Bielefeld Conspiracy! Non-believers can engage in many of the same type of arguments for the non-existence of deity as well as Bielefeld. Maybe all the believers are in on some big trick against nonbelievers? Perhaps the miracles never happened! Pictures and other evidence can be falsified. The list goes on. (Also consider Clark's third law: "Sufficiently advanced technology is indistinguishable from magic." Maybe the believers or other miracle-workers used technology!)
A deity must not do much to leave room for doubt; simply giving free will to its creations, working within natural law, and not publicly revealing itself may be enough.