Yup, that's still sounding a lot like Buddhist meditation to me. There, too, your mind isn't supposed to be quiescent, in fact one of the early pitfalls is "dullness", or simply shutting your brain off. You're meant to remain perfectly focused, and entirely alert. "Tenere", while probably not the way Buddhists would describe their meditative practice, isn't at all at odds with it.
I'm not an expert, but it feels to me like in both traditions the goal is to be completely present, ie all mental faculties up and running and brought to bear. Perhaps for the Christian monks there would be an added element of communication with God (but, perhaps, it was simply being in His presence?), whereas for Buddhists there isn't... I don't know.
The point of Buddhist meditation isn't to somehow beat your thoughts into submission. It's first to acknowledge them, as you say allowing them to come and go. During the course of that you realize that they are something separate and distinct from your self (further realizations about the self come later, of course). Later on, as you progress, they eventually don't intrude anymore. It's not that you've defeated them, they simply don't pop up.