Not really an expert on guns but I would have thought the rifle was better suited, I think (especially with the sites), it would be more accurate and a shot gun would would damage the property - given that he was 'supposed' to be shooting them in the yard.
What is odd, was that he initially said he had shit the rifle then changed his mind (like you could forget), guess he realised there was no physical evidence of such shots being that he made up the story about the rabbits.
It's unclear whether he initially said he fired the rifle or not. He said he didn't get the rabbits. That could be because they got away without him getting off a shot or firing but missing.
In his initial account could simply have been vague and misinterpreted. He could have been vague on purpose to keep his options open. Later he was very specific in saying they got away because they were gone by the time he got out there. The police didn't quote him verbatim to say he fired and missed but rather that he missed them which they interpreted as him firing and missing.
Since Jeremy made up the whole account he hadn't really thought out the details. He didn't clearly think about what to say about exactly how he came across the rabbits and thus his first tale where he spoke about being in the kitchen with the family and running after the from the kitchen. He fleshed it out more later as he thought in detail what to make up. He wanted police to believe a fully loaded gun was left for Sheila to grab so it is unlikely he originally intended for police to think she fired but missed. It seems he just didn't fully flesh things out and didn't realize they would think he meant he fired and missed when he said they got away.
None of the cops at the scene say he specifically said he fired and missed butt rather simply he missed them which they interpreted as him firing. In many ways this is like the bed issue with far side of the bed and near side of the bed meaning the floor next to the bed not actually the bed itself.
That is not to say Jeremy didn't drastically change some things. Jeremy wanted the police at the scene to think Sheila could use all the guns and thus made up teaching her to use them. He realized the detectives would ask why he would teach someone he considered insane and didn't like or trust so when speaking to them he told a different account.
It is not impossible Jeremy said he fired but missed to the initial responders but later thought the better of it and decided it was better to pretend he fired no shots so the gun could be fully loaded for sheila to find. Still I think the entire plan of making up the rabbit story was to suggest she found a fully loaded weapon of opportunity and thus all along he would recognize the value in claiming the rabbits left before he could fire. I believe the police and family for that matter simply misinterpreted.
Addendum:
Upon review of what the initial responders said he told them the magazine was full when he left it in the kitchen. Quite clearly he was not suggesting he fired and missed since he was suggesting the magazine was full.
The family kept saying he said he missed the rabbits, they seem to be the main ones who misinterpreted. The detectives were unsure whether he meant missed as in fired and missed or didn't fire so they made Jeremy clarify which. In Jeremy's August 7 written statement he said he didn't fire any shots because they asked him clarify.