IANAL, but I have had to deal with trademark issues and as I understand trademark law you would be considered to be in exactly the same market. The difference between, say, pet food and hosting would be considered relevant, not between storing backups and storing PHP websites.

Also, regardless of whether you did so or not, the logo really does look like a slightly masked copy. Do an image search for "blaze" to see a number of possible treatments on that theme that would not result in such a resemblance.

I am less sure about the name, but the difference is swapping "b" with "st", so I wouldn't be surprised if that would also be considered trademark infringement.

Anyway, it depends how you understand "not a problem"; You might feel morally justified, but I believe that legally you are on pretty shaky grounds. On the other hand, Backblaze would have to kick up a fuss for that to actually matter.