I’m not sure the linguistic differences here are as cut and dried as you would like them to be. Estimate and approximate are both verbs, so you can derive nouns from them both for the process of doing the thing, and for the thing that results from such a process.
Estimation is the process of estimating. It produces an estimate.
Approximation is the process of approximating. It produces an approximation.
You can also derive adjectives from the verbs as well.
An estimate is an estimated value.
An approximation is an approximate value.
But you’re right that the ‘approximate’ terms make claims about the result - that it is in some way near to the correct value - while the
‘estimate’ derived terms all make a claim about the process that produced the result (ie that it was based on data that is known to be incomplete, uncertain, or approximate)
Estimation is the process of estimating. It produces an estimate.
Approximation is the process of approximating. It produces an approximation.
You can also derive adjectives from the verbs as well.
An estimate is an estimated value.
An approximation is an approximate value.
But you’re right that the ‘approximate’ terms make claims about the result - that it is in some way near to the correct value - while the ‘estimate’ derived terms all make a claim about the process that produced the result (ie that it was based on data that is known to be incomplete, uncertain, or approximate)