Chert, flint and agate are all quartz, but are sedimentary varieties of it, with quartz itself being an igneous mineral. The rock in question looks to be a chert to me. This is evident due to the glassy texture, lack of visible crystal structure and the conchoidal (rounded) fractures which are very indicative of quartz.
Its hard to tell from the photo, but the darker one could possibly be an andesite or gabbro, with associated quartz or a calcite interior. Id have to see it n person to give a definitive answer though.
The purple one is amethyst (quartz).
HH