Yes, you have to execute the php command directly on the server. If you do not have command line access to the server, you cannot do that. In that case you could try using your browser to use the web updater in mw-config/index.php instead.
However, I would not recommend this. Downgrades generally are not supported.
There are several reasons. E.g. while doing an upgrade changes to the database are done, e.g. content is moved from one column to another or content in a column is modified in a way. However, there are no scripts in older releases, which could undo these changes. So the upgraded DB will not work properly with an older version of MediaWiki. If you want to downgrade, you should do that by taking an old backup, in which you have only used the old version. Using an old MediaWiki version with a newer database will break things (just like happened in your case). Another problem in which you are likely to run after having done a downgrade is that a new upgrate (to in your case MediaWiki 1.19) will not work correctly: While upgrading, the updater checks, if some columns in the DB are there. If they are not, some update routine is executed. This routine can also populate columns with content or change existing content. If however you have a DB, which was already updated some time before, but then used with an older version of MediaWiki, these updates will no longer take place and you will end up with missing/wrong content, which was inserted (or not inserted) by the old MediaWiki version, which you downgraded to.
Bottomline is: Take a database backup from before your downgrade and it will work with 1.19 or take an older database backup from MediaWiki 1.16 and it will work with 1.16. Everything else might -at least in the long run- leave you with a broken system.