When a trader puts in a buy order to be filled at the current offer he is said to be "taking". Also he is never eligible for a rebate if filled on the offer.
Some NYSE traders who strongly want a fill will enter their buy order a penny or two above the current offer because by the rules they must be given the best available price-- they are "taking liquidity" as well.
(When it comes to definitions, I try not to roll my own.)
This is not necessarily true.
If I use an inverse exchange,, I will be provided a rebate for taking liquidity (entering via market order). Alternatively, I could provide liquidity and use a traditional exchange to collect a rebate also.
Orders that are entered at an aggressive price (better than or worse than the nbbo) do not get any priority with respect to their position in the queue, so I don't know where you were going with that second example.