>> if a column is of type INTEGER and you try to insert a string into that column, SQLite will attempt to convert the string into an integer. If it can, it inserts the integer instead.
There's nuance to your quote. From my recollection, this means that "321a" will be inserted as "321", but "foo" will be inserted as "foo" (into an INTEGER column). Definitely a wart, on an otherwise fantastic system.
The expression "CAST('321a' AS INTEGER)" will do as you suggest and ignore the trailing 'a' character, yielding an integer 123 result. But that only happens for an explicit CAST. Automatic type conversions must be reversible. That means that '321a' is inserted as a string in an INTEGER column, but '321' (without the trailing 'a') will be converted into an integer 123.
PostgreSQL, MySQL, and SQL Server do exactly the same thing for the '321' case. For the '321a' case, the other three throw an error whereas SQLite just cancels the type conversion and inserts the original string.
Either way, the fact that you can end up with strings in an Integer column is certainly surprising...
sqlite> create table test (foo INTEGER);
sqlite> insert into test (foo) values (123);
sqlite> insert into test (foo) values ("blah");
sqlite> insert into test (foo) values ("123a");
sqlite> select * from test;
123
blah
123a