I just asked the Taipei tax office clerk about this, and according to her the calculation formula should only use the number of resident days when it’s less than 300. Above that, the multiplier should just be 365/365.
She also said that they’d correct that later if the software had applied the formula incorrectly, so consider yourself warned @qwert_zuiop 
I hope they’ll at least void the penalty if their own software is doing the calculations wrong… And I’ll also need to pay special attention that they will also adjust the deductions and not only the income (both were automatically adjusted by the software…)
Incidentally, the lady in the tax office also had no idea what those fields in the offline software were referring to when I showed her the screenshot in my above post:
It’s this kind of thing that makes me unsure of the correctness of info given by the tax office sometimes when it comes to less conventional situations like ours (despite them mostly being helpful and friendly).
I did try asking her about the thread topic (work done by Taiwan residents while physically outside Taiwan) while I was there. This was another question she wasn’t able to give a good answer to and the conversation devolved into some frustrating back-and-forth where we were both talking about different things, but the impression I got was that (i) it would be possible in principle to disregard income earned while physically outside Taiwan, but it would require additional evidence that the income was actually earned and/or paid while outside Taiwan (e.g., entry/exit records, documents from overseas tax authorities, possibly notarized stuff) and (ii) this would be easier in the 183–300 day range to meet the <300 days requirement mentioned above.
No definitive answer, though – we were both getting a bit irritated with the conversation at that point and it didn’t make enough difference to me to want to pursue it. (She was also looking at it from the perspective of someone with a full-time employer, and she didn’t seem to grasp how somebody might work remotely for, say, a couple of afternoons while on “vacation” or outside Taiwan for two months.)
Yeah, that’s why I am filing online - at least I am in control what I write down…
In my case, it’ll probably end similar to last year when I ended up going to the tax office after filing online to straighten out everything…
Just remembered - another thing she asked me about was why my foreign currency amounts converted into TWD at the tax office rates didn’t exactly match what the filing software had calculated. Mine were slightly higher than those calculated by the filing software, which meant I paid slightly less tax than I should have done… because the filing software ignores decimal places in foreign currencies and just takes the whole number.
I told her I’d give her the missing few TWD if she wants, but like… that’s not my fault, it’s your software. 
Everyone deserves a little break
It’s better in your pocket…