Does a surveying office do other more precise measurements?
Yes.
How much does such a thing cost?
Which thing exactly do you mean? If you can do something with coordinates, they should be provided to you free of charge. Calculating four right-angled measurements should also be a service, with the expectation that you commission the site plan and/or the building survey. A penny-pincher might charge about €100.
How does an architect proceed in this case?
An architect obtains the coordinates—in the ideal case, asks about their accuracy or is informed by the surveyor—and imports them into his CAD software. This way, he obtains any desired measurements at any point within the accuracy limits of the measurements.
The note
"Span dimensions are determined from coordinates and must be verified on site!"
indicates that the planned parceling has so far only taken place on the computer and slight changes may still occur through on-site surveying. These usually happen below 5 cm. If you want to be absolutely sure, you have to wait until the cadastral office has recorded the new boundaries in the land registry.