Re: Reading "Approved" back
What you are doing is correct, but the ATCO is not, as you have identified.
The difference between 'approved' and 'cleared' is important: If you are cleared to do domething by ATC, they are taking on the responsibility for nothing getting in the way of that permission: Cleared for take-off, cleared to transit etc.
If ATC approve something, they are giving their permission but not necessarily removing the responsibility from the pilot to take measures for it to happen safely.
An example would be 'Push approved'. If you were to take this as a clearance and push back straight into a catering truck, that would be your responsibility, not ATC's.