Letter writer snapped at my reminder email. What can I do in the future to prevent this, and should I still accept his letter?
How can I prevent this from happening in the future?
Well, you can always go live on a desert island, or join an order of silent monks (and even living among monks would not guarantee that you will escape such foibles of human nature).
More seriously: you can’t prevent it, and you shouldn’t try either. You did absolutely nothing wrong by sending a reminder email. For every professor who responds irritably to your emails there are at least ten others who would appreciate your reminder email, and for some of them the reminder would even be needed to get them to upload their letter. Thus, you stand to lose much more than you will gain by changing your behavior because of the professor’s response. Just accept that working with other people necessitates occasionally having some strange and perhaps unpleasant interactions (trust me, you will have many more), and move on.
By the way, I also think it’s highly unlikely that the professor will modify his recommendation because of your email — that would be an extremely unprofessional and unethical thing to do, much more so than replying irritably to an email. Anyway, good luck with your applications.
I feel that I have said or done something wrong. How can I prevent this from happening in the future?
The only odd thing you did was remind via both the online system and the personal email without acknowledging you did so in the email ("I am writing you an email on top of the online notices because I was told those sometimes get caught in spam.")
You could follow up with: "My apologies. I wanted to put the links at the top of your inbox so you could find them easily. I thought I should follow up with an email because I was told these can sometimes go to spam. I realize now this could be a bit obnoxious. Thanks again for your willingness to write a letter." You might stop by later and find out if he doesn't want ANY reminders or if he just wants a single one from the online system or a single one via personal email.
I am now worried about the strength of this professor's letter. I have four letter writers, and I am tempted to drop his letter on some January schools.
I would say you need to figure out if his behavior with you is just his typical behavior, or if he acts more negatively toward you than toward other students in your position. It seems most relevant to figure out your position relative to HIS baseline, not relative to a "nice person."
This is just a reminder that first due dates are tomorrow: [list of a couple of schools]. Hope you have time to write still. If you haven't received an email from these schools on instructions let me know.
This wasn't worded very well. By explicitly declaring your message a "reminder", you've:
Taken a role in managing their activities.
Suggested that you have less than perfect confidence in their reliability.
In general, both of these things might inspire offense. The primary issue is that, as a student requesting a recommendation letter, you're making a request of a professional-superior; deference and thanks are the typical expectations.
So, you'd have two ways to approach this: either as a professional-peer trying to coordinate an activity or a professional-subordinate requesting guidance from a superior.
To word something like this as a professional-peer:
I'm working on finalizing my applications to [list of schools] that have their deadlines tomorrow, so I just wanted to touch base with you on the recommendation letters and thank you again for your time and consideration. Please let me know if they failed to send you the invitation via email or if there's anything else that I could do on my end.
To word something like this as a professional-subordinate:
Thank you for helping me with the application process with your recommendation. Since a few of the deadlines are tomorrow, I'm trying to make sure that I got everything in order, including ensuring that I properly provided those kind enough to provide recommendation letters with access to the applications. Please let me know if there's anything else that I need to do, and thank you for your time and consideration!
Whether you approach a recommendation-writer as a peer or subordinate will depend on a lot of personal factors, so that's a case-by-case thing.
Stuff to avoid:
"Reminding" them. Reminding someone suggests that you're not 100% confident that they were on-track. However plausible or reasonable this might be, some might still take offense to it.
Managing their activities. In some sense, they've agreed to do something for you, and thus you might reasonably argue that they're obligated to perform the agreed-upon service. And while this might seem technically true, it'll often rub people the wrong way; it's best to avoid sounding like a micro-manager.