Until reaching the threshold of 20000 reputation points, during the review process I was presented with the option to "Recommend Deletion". Now that I have reached it, this has turned into "Delete". Furthermore, after reaching the 10000 threshold I got access to the "review > Tools" page, where again I was able to cast delete votes. How do these three delete options differ from each other?
There are two ways how the community can delete answers (well, actually, it can also via spam flags etc. but this is orthogonal):
Users with 20k+ can vote to delete an answer (provided its score is negative). This can be done on the answer directly using the link below it. Three such votes will delete the answer. Technically, this is very similar to voting to delete a question (which is a 10k+ privilege).
In the Low Quality review users can suggest to delete an answer. For users below 20k (or, as Daniel pointed out, users above 20k that are out of delete-votes for the day) this will be recorded as "recommend deletion" for 20k+ users an actual delete vote will be cast, too. Six such suggestions (recommendation or actual vote) will delete the answer.
That there is a difference in the LQ queue depending on below or above 20k explains the change you saw.
The "tools" page just aggregates information related to the first type of delete-voting, showing the current actual delete votes. It can be used as a gateway to posts one might want to cast a delete-vote on, yet one never literally votes from the tools page.
The deletions via votes and from review are slightly different (which type it was is also marked on the post).
A deletion from review can be undone by OP voting to undelete. To undo a deletion by votes by contrast needs three votes (that can include OP).
The criteria for which answers are eligible for deletion are slightly different. For review it is non-positive score, for votes it is negative score.
To put it somewhat differently, a users can recommend deletion in LQ review and six such recommendations delete, a user with 20k+ can vote to delete answers and three votes delete. And, for a 20k+ user doing the former the system will automatically do the latter (except if they are out of votes).