"Transaction downloaded by" and "Transaction downloaded on"
These fields record when the Online Express system sends that transaction to the download plug-in. (It does not refer to when a transaction is sent to Raiser's Edge batch.)

The noted user does not have to open the Download Transactions window for their name to appear there. It means that they happened to be the user who opened Online Express (clicking on Online Express in the left menu) when that transaction was synced from the Online Express servers to your Raiser's Edge.

For example, say a user needs to work in Email Marketing but they last closed out of Online Express on Donations (by clicking on anything else in the left Raiser's Edge menu). Online Express remembers the last area that a user works in, so this user's Online Express will open to Donations the next time they click on Online Express in the left Raiser's Edge menu. And in that next visit if a new donation transaction is available to be downloaded to the "Download Transactions" tool then, that user will be "Transaction downloaded by" user in the download because Online Express and your Raiser's Edge synced at that time.

This "download" information does not mean when this transaction is moved to a Batch. A transaction will no longer appear in the list of the download plug-in when it's moved successfully to a gift batch.

"Transaction currently checked out to" and "Transaction has been checked out since"
These fields show the user and when they opened that transaction to work on it, such as linking constituents and funds. When a transaction is "checked out", it's important to complete that as soon as possible to process it to a batch so the gift can be committed and recorded. If a transaction is checked out to a user, other users can see it but they cannot edit it / process it.