The answer to the previous question [previous question][1] included a link to [this question and answer][1], answer][2], which explains it, specifically where the answer says:
If you suspect that the tile was not correctly expired, there is a way to manually overwrite this expiry mechanism by appending a /dirty to the url which will append the tile to the back of the rendering queue. However, in nearly all cases this should not be necessary.
So if you follow your link, right click on the tile that you want to dirty, you'll get a URL like
[http://a.tile.openstreetmap.org/15/9543/14534.png][2]
[http://a.tile.openstreetmap.org/15/9543/14534.png][3]
If you append /dirty to that you get
[a.tile.openstreetmap.org/15/9543/14534.png/dirty][3]
[a.tile.openstreetmap.org/15/9543/14534.png/dirty][4]
and you'll get a message "Tile submitted for rendering".
You can also do
[a.tile.openstreetmap.org/15/9543/14534.png/status][4]
[a.tile.openstreetmap.org/15/9543/14534.png/status][5]
to see what the current status is.
You'll also notice that that particular tile already displays "Sherman Avenue" because I've dirtied it manually while answering this question.
[1]: https://help.openstreetmap.org/questions/16021/street-name-change
[2]: https://help.openstreetmap.org/questions/178/how-often-does-the-main-mapnik-map-get-updated
[2]: [3]: http://a.tile.openstreetmap.org/15/9543/14534.png
[3]: [4]: http://a.tile.openstreetmap.org/15/9543/14534.png/dirty
[4]: [5]: http://a.tile.openstreetmap.org/15/9543/14534.png/status
