You’re not a dick and you weren’t being a dick, I totally agree about writing on the internet - it’s tricky to feel and to tell, and I imagine I must be insufferable at times.
I can see your point now and I think we might agree on more than either of us first thought. I can see how much you care for your family in every word you’ve written, and I’d be a bit surprised if one of your kids was at home, told you it’s their birthday, and your reaction would be to laugh about it, and then carry on with your day as normal with no further reaction or acknowledgement of it.
So - Ii’s not about remembering, or about the card or presents or cake: it’s what you do with the information in front of you. From the sounds of it you’re demonstrating your care brilliantly every day.
(I have the same type of brain wiring you’ve described, incidentally. My extremely accepting wife would ask me to buy her some flowers, nothing showy, just something to brighten up the house and remind her of me. And I’d hear that as “she wants me to remember to buy her flowers sometime” and file it away. And several years later we figured out that gap in our communications and nowadays, when it’s important enough to her that she asks me to buy her flowers - I buy the flowers. Turns out that that was all it took to demonstrate that I care.)