Allow me to sign in to Trello with my personal account.

[Mike Ruhlin]

I have an existing Trello personal account.

I also have a Jira account that I use for work.

On my work computer, I have followed a bookmark to one of my personal Trello boards.

I'm presented with this login screen:

There should be a "use a different account" button on this screen.  "Never Mind" takes me back to the trello home screen, and I'm afraid of what details the "Create your account' button is going to share with my company.

[Gaurav Kataria]

You can be logged into two separate Atlassian accounts at the same time (let's say one account where you use Trello for your personal work and another one where you use Jira for your company work). However, you will need to switch between those accounts when accessing Trello or Jira. See this page https://support.atlassian.com/atlassian-account/docs/switch-between-multiple-atlassian-accounts/