every time the rate is reported it is counting transactions. But you are right, technically when talking about "throughput" we should count payments not transactions and one tx can contain multiple payments.
I don't feel like checking the data (it's a lot of work), but my gut feeling tells me the number of transactions per block is going
down, despite Segwit, because of the batching which leads to much larger transactions. Counting transactions as a measure of Bitcoin's scaling would give the wrong idea.
Especially since there are a variable number of transactions in each block, so for any given time period, you would have to analyze the last 2016 (for example) blocks and take their average transaction count to compute the TPS more accurately.