The shaking hands on the first injection is universal - almost everyone describes it that way. The button mechanism on auto-injectors is counter-intuitive until it's not, and then you wonder why it felt hard. Most people are completely relaxed by shot three or four.
The shaking-hands first injection is everyone's experience - it gets mechanical very quickly. Days 2-4 are when the GI adjustment arrives for most people.
Two hours post-first-shot is exactly the right time to be excited and nervous simultaneously. T1D adjustments on top of everything else is a real consideration for the first few weeks - how's glucose tracking since the injection? Two hours in is still the very beginning of the pharmacokinetic curve.
Kinda terrified at 5 weeks in on the first shot despite tons of research is the universal experience - the research doesn't make the first needle less real. The OP's hands shaking while doing it is the most honest version of this moment. Week 5 is usually when the whole thing starts to feel normal. How did the side effects track through those first weeks?
The spring-loaded auto-injector is the tool that removes the psychological barrier for people who can draw a syringe correctly but can't push it through their own skin - the button-press mechanism handles the moment that most people freeze at. The Owen Mumford version has a good track record in the community for consistent injection depth and speed; worth having around even if you get comfortable with manual injection, because it's genuinely faster