I am a Surveyor with over 20 years of experience with GPS and coordinates systems, hopefully I can explain a little for you. Coordinate systems can difficult to explain in couple paragraphs.
Your surveyor is asking if you want in ground or grid, because ground is what is used in the (real world) if you were in the field measure with a tape . Grid is a measurement from a projection which will have different scale factors applied to your measurements, depending on where you are with in the projection. So if you are using your data for design and you are trying to measure in the real world ground distance, you will want your output coordinate system to a coordinate system that is at ground or real close with small amounts of distortion, like a LDP (low distortion projection).
Most of your out of the box coordinate systems like State Plane, UTM and etc… are for large areas covering many hundreds of kilometers. Which is good for planning over large areas. So if you are exporting to a UTM coordinate system and you measure on the screen a 100 feet, that is not 100 feet like you would measure on the ground with a tape measure. You would have to figure out what the scale factor is and apply to get your ground measurement.
LDP (low distortion projection) are being used a lot for smaller area like a city or county wide coordinate system. Which closer to ground measurements and almost eliminate a need to scale measurements, So that you can mix gps measurement and terrestrial measurement with out much difference, unlike if you are trying to mix UTM coordinates and Terrestrial measurements.
So if the city is your working in has a local coordinate system using a LDP, then I would use that as your output coordinate system. Pix4d will let you choose a prj coordinate system file. A prj file is an text file that give all the coordinate system variables, which can be created in third party software, like Esri and etc… or get from someone that has all ready created the file.
If there is not one available in your area, you can create your own or use PIX4D site calibration options so that you can calibrate to ground coordinates.
Might want to contact your surveyor and ask for his assistance. Show him your options in PIX4D and they should be able to help you.
Good Luck and have a good day.