What formula can I use to calculate the salinity percent in a sample of water that I am adding salt to? Is is salinity = salt mass / total mass?

There are several common ways chemists calculate concentrations:

molarity

This is the concentration in moles per liter. A certain number of moles is placed in a flask (either by weight or by volume from a solution of known concentration), and the sample is diluted to a certain volume. The number of moles divided by the number of liters is called the molarity.

This is the most commonly-used type of concentration specification in chemistry.

weight/volume percentage

A sample of known mass is diluted to a particular volume. 100% is defined as 1 gram of solute per milliliter of solvent, so (for example) 5g diluted to 100 mL gives a 5% solution.

Precise workers will call this sort of solution a w/v solution.

molality

This is much less commonly used, but is a handy unit in some situations (typically when we don't have volumetric glassware of the proper size!). In this type of solution, a certain number of moles of solute is dissolved in a certain number of kilograms of solvent, so (for example), 0.01 mol of NaCl dissolved in 10g of water is a (0.01 mol/0.01 kg) = 1 molal solution.

weight/weight percentage

This is analogous to molality, and is usually used when preparing mixtures of two solutes in a solvent. w/w percentage is defined as grams of substance A divided by grams of substance B, times 100%. For example, a photopolymerization experiment I ran the other day had the monomer and the initiator both dissolved in water; the initiator was present as 1.5% w/w with respect to the monomer: I used 0.015 g initiator for every gram of monomer.

You can also prepare solutions w/w with respect to solute and solvent, but that's much less useful than w/v.

Here's how it works, bottom line: most solutions are prepared so that a certain volume will deliver a certain amount of solute. That way they're the most convenient to work with. Therefore we typically prepare solutions either by molarity or by w/v percentage.

To get back to your question, for chemists salinity is typically quoted as grams of salt per 100 mL of water: that's the way w/v percentage is defined. BUT... them medical types just have to be different. They insist on quoting salinity as a w/w percentage, grams of salt per grams of water. See this previous answer.