CREATE INDEX vs ALTER TABLE ADD INDEX - MySQLism, or SQL Standard?

I don't think that the SQL standard defines how to create indexes at all.

A quote from this Wikipedia page:

Standardization

There is no standard about creating indexes because the ISO SQL Standard does not cover physical aspects. Indexes are one of the physical parts of database conception among others like storage (tablespace or filegroups). RDBMS vendors all give a CREATE INDEX syntax with some specific options which depends on functionalities they provide to customers.

The Postgres manual seems to support this here:

There are no provisions for indexes in the SQL standard.

More evidence under this related question on SO.