My understanding is that when you are describing the experience gained from your previous employment, an uncountable form of the word 'experience' without any article and no plural form should be used, but sometimes I read cases where my presumption doesn't seem to hold the water. Was I wrong?

Teaching experiences: List your teaching experiences
It is usually uncountable in the context you describe, but there is nothing to prohibit the countable version, really. In the example you give, the verb 'list' lends itself to the countable form.
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SystemAdministrator: A system administrator takes care of the inner workings of the entire system. These users have the ability to promote, ban and modify other users.Teachers: Users in this role are certified teachers. This may include DELTA, CELTA, TESOL, TEFL qualified professionals. Email a scan of your qualification to an admin, if you wish to be considered.
No, it should be "list your teaching experience" if they mean employment history. Teaching experiences would be specific things that happened in the classroom.