Did I understand the definition of "gross primary production" correctly?
Yes, you more or less got it!

If you have only 3 primary producers (say 3 plants) in a given ecosystem.


Plant A produces 2g of organic carbon per second
Plant B produces 5g of organic carbon per second
Plant C produces 1g of organic carbon per second


, then the GPP is 8g per second.

This produced organic carbon can be in the form of leaves, root or whatever tissue these plants are growing.

Note: the numbers chosen are irealistic both in terms of production per second and in terms of number of plants in an ecosystem.


  gC (grams of carbon assimilated) m-2 (m squared) year-1", what is meant by "grams of carbon assimilated" ? 


The whole point of primary producers is to fix carbon. That is to take inorganic carbon (such as $CO_2$) and make it organic (such as $C_6H_{12}O_6$ aka glucose). This is why we are typically particularly interested in carbon and not so much in water.

For more info, you should have a look at an intro course to respiration and calvin cycle.

Note that if you remove water from an organism you are roughly speaking left with 75% of carbon, 10% of Nitrogen and the rest made of different elements.