what does contemporary mean

“Twist” can mean several different things here, from just a “change” to a “surprise”.
A contemporary twist is a ‘modern angle’, ‘modern slant’.

We do not care for their “ideals” expounded in contemporary histories and eulogies.
I have become a contemporary of David and the patriarchs of Israel.

We will use the City of Southampton and surrounding areas to make a practical exploration of key topics related to Britishness. By considering ideas of belonging and identity, you will discuss and experience a range of British attitudes and practices related to subjects such as: the nations and nationalisms of Britain; (multi)cultural diversity; language and citizenship, social inclusion, and the British class system.
‘What does it mean to be British?’

References:

http://www.thesaurus.com/browse/contemporary
http://www.southampton.ac.uk/international_summer_school/themes/modern-britishness.page
http://libanswer.pstcc.edu/faq/170590