MIN/MAX aggregation over an indexed column may return wrong result

For a DB Table having a id field which is the table’s auto-incremented primary key
SQL-query:
SELECT max(id) FROM _between_bug2 WHERE id < 303 and (id BETWEEN 267 AND 287)

gives the result:
max(id)
267
, which is incorrect as id BETWEEN 267 AND 287 should be equivalent to (id >= 267 AND id <= 287 and the max id value that satisfies the condition is 287, not 267

At the same time SQL-query
SELECT max(id) FROM _between_bug2 WHERE id < 303 and (id >= 267 AND id <= 287)
max(id)

gives the correct result:
max(id)
287

here is a minimal example to create DB table and reproduce the issue

CREATE TABLE _between_bug2 (id int(13) auto_increment primary key);
INSERT INTO _between_bug2 (id) VALUES (267);
INSERT INTO _between_bug2 (id) VALUES (287);
INSERT INTO _between_bug2 (id) VALUES (303);

SELECT max(id) FROM _between_bug2 WHERE id < 303 and (id BETWEEN 267 AND 287)

and here is an SQL fiddle to reproduce it