You are correct. When you buy with a market order you are buying at the offer (or ask), when you sell with a market order you are selling the bid. People tend to talk about passive buyers and aggressive buyers. The market order is aggressive because you want to buy straight away so pay the higher offer price. But a more passive buyer might try to buy the bid with a limit order but risks not getting filled. The T&S shows the 'aggressor' so it will show buying at the offer in green and selling on the bid in red as you say.
As an example. Watch the tape long enough and you will see an iceberg order. Price moves up, green on the tape, then a large amount of buying goes through at one price, an amount that was larger than the amount on the Depth of Market (DOM), that was at that level waiting to sell. This means there was a hidden iceberg order there. So although the tape is green and shows a large amount of buying it was because a large passive seller had a resting order there. Depending on the location of this order it can sometimes give buyers pause for thought because they don't know how much more somebody is looking to sell there and the market might drift down as the late buyers get out to reconsider their position. Or a more aggressive buyer carries on pushing, and the seller has finished their business at that level, so price carries on through.