I think the defense can also request a judge to add LIOs, arguing that they are supported by the evidence. But I think that's less common: if the defense doesn't believe the DA has met its burden on the higher count, why give the jury an option to convict for something else? But I guess there are cases where the defense feels the jury is going to convict a really unlikable defendant of something, so it's a calculated gamble to give jurors a lesser option.
Conventional wisdom holds that juries have a tendency to "split the difference" if you let them, so often one side or the other will oppose LIOs in order to push the jury toward the desired result. You're right that it's often a matter of strategy, at least on Law and Order. LOL.
Note: I'm not sure everyone realizes how hotly jury instructions are contested. But as someone who has had to type and retype them, I can attest that the instructions to the jury (including on matters such as LIOs) is often a result of fierce battling between opposing counsels under the supervision of the judge.
