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If you look at this article which describes The Noble Eightfold Path,
Chapter VI: Right Mindfulness (Samma Sati) talks about "bare attention"
Chapter VII: Right Concentration (Samma Samadhi) talks about "centering the mind on an object"
So I think they're both Buddhist, but one is called "mindfulness" and the other "concentration".
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“And what is the faculty of sati? There is the case where a monk, a disciple of the noble ones, is mindful, highly meticulous, remembering & able to call to mind even things that were done & said long ago. (And here begins the satipatthana formula:) He remains focused on the body in & of itself — ardent, alert, & mindful — putting aside greed & distress with reference to the world. He remains focused on feelings in & of themselves... the mind in & of itself... mental qualities in & of themselves — ardent, alert, & mindful — putting aside greed & distress with reference to the world.”
SN 48.10
You could give a read to Mindfulness Defined by Thanissaro Bhikkhu at http://www.accesstoinsight.org/lib/authors/thanissaro/mindfulnessdefined.html