Do yourself a favour, and bill by the day, rather than by the hour. Engineering is creative work, and you're paid to think - code is just the end product. Thinking happens consciously and unconsciously, at all times of the day. Billing by the day will allow you to focus on the work rather than the administrivia.
Whatever you do, do not drop your rate: you'll just be undervalued by your clients.
There is a science component to ideas in the shower. I read somewhere that the shower puts you into a similar state to the waking/sleeping transition that people like Einstein tried to ride by sitting and holding something heavy and loud, so if the slipped too far towards sleep they'd drop it and wake themselves up. I don't have any references to where I heard that though.
Similar lines, I read about some earlish computer pioneer that was offered a senior position at a company and the deal breaker was whether he could get a shower in his private bathroom off his private office. Company said no so he moved on. "I have my best ideas in the shower." I don't recall who it was, Cray sticks in my head, but I don't think it was him, someone similar though.
I have also heard it referred to it as "body is busy, mind is open" states. In the shower you've got a routine that keeps your body busy enough doing things, but you generally don't have to devote a lot of mental effort to that routine, most of it is truly muscle memory.
Similarly, things like going for a walk or getting a little exercise can give your body plenty of things to do, but fewer things the mind needs to check on.
Alan Kay has often said that he gets his best ideas in the shower (I think the idea for efficiently implementing overlapping windows was one of them). I can't remember where I read it but he asked for a shower in his office at PARC but they said no (but he nevertheless stayed on).
The issue with daily rates is that the client assumes you are available/working for a "full" magical 8 hours. On my side, I don't want to bill for a "full" day if I did some quick 1-2 hours of work. I also don't want to do such 1-2 hours of work for free.
> Engineering is creative work, and you're paid to think - code is just the end product.
No you aren't. No one cares if you think. They care about what thinking is (supposed to be) getting them. It just so happens, though, that whatever that is requires thinking. So you do charge for that. But no one is paying you "to think".
> If you are a manager, an architect or a scientist, all you do is think. You literally don't do anything else.
First of all, no; it's not true that, "You literally don't do anything else". Even if we were to accept that, though, it's besides the point, because what you're saying right now is a change of subject.
What I said was that you don't get paid to think. You're getting paid for what the customer (hopes) they'll get from you. (And yes, that's true even if they never get anything from you.)
To say it again: nobody cares if you're thinking. The exercise of thinking has no intrinsic value to the people who people who are paying you. All of the value is consequential. Ergo, even if you are thinking over the course of your work day, you are not really getting paid to think. If you could somehow manage to deliver the same results without thinking, you'd still get paid--because you're not actually getting paid "to think".
> If you could somehow manage to deliver the same results without thinking, you'd still get paid--because you're not actually getting paid "to think".
Reductio Ad absurdum:
If you are a miner, you are not being paid to dig, you are being paid to find coal. If you could deliver coal without digging, you would still get paid.
But wait, you are not being paid for coal, you are being paid for keeping house warm. Id you could keep their house warm without them burning coal, you'd still get paid.
But wait, they don't actually want to keep the house warm, they want to not feel cold. If they could achieve that without heating the house, you would still get paid.
So no matter what you job is, I couls always say 'you are not getting paid to code/trade/do X, instead people want results of X' and you could repeat that process for X,Y and Z ad infinum. So the argument has no value
I'll rephrase: they're not paying you to mindlessly churn out lines of code. The end product matters. But the bulk of that time is spent thinking, whether you realise it or not, and frankly that's a much more useful perspective for the OP to take, as they're trying to reconcile their comfort with charging for time in meetings and lunch, because they think they're not being "productive".
Whatever you do, do not drop your rate: you'll just be undervalued by your clients.