None, nor is that the issue here. It appeared to the officer that OP was using the private property to bypass a traffic control device. The fact OP was apparently only on the property long enough to quickly check the air (I wonder if this required any maneuvering as that might bolster his version) and then immediately left the property. It becomes a matter of OP's word against a law enforcement officer's. Courts tend to give more credibility to the officer as they are considered an unbiased party.
It's not a cheap shot to point out that this isn't CA. What is legal or not in CA has no bearing on this as it took place in Illinois.
Also, while it may not be illegal statewide in CA, municipalities can and do enact ordinances for this type of thing.
Obviously it is illegal in Illinois. Ejay even quoted the law in question, which you could have googled yourself since OP lists it as well.
Side streets is fine. Cutting through private property is illegal. The text of the statute is
"(b) It is unlawful for any person to leave the roadway and travel across private property to avoid an official traffic control device."
It's not a cheap shot, it's the Illinois statute we're talking about.
California doesn't have such a law, though they you'd be well advised when doing this that you are not trespassing and that you follow the rules for stopping and yielding before reentering the highway.
The officer can't know, much less prove, that Denise drove into the gas station "to avoid an official traffic control device." This is because people can't read other people's minds beyond a reasonable doubt. This is very different from, say, whether she made a complete stop at a stop sign, or whether she was speeding. As long as the O/P can testify under oath to another plausible reason (which she did give), the charge should be dismissed. Even in traffic court.
No, that is simply wrong. When the court hears two competing versions of events, the court is entitled to look at all of the available evidence and determine which version is more credible. A defendant's denial of intent or other excuse-making is not a basis for automatic acquittal.