The latitude on maps is a third kind, called geographic or geodetic latitude. It uses the perpendicular from a mathematical surface that approximates the shape of the earth extremely well. (Whenever you have the adejativ geodetic applied, it usally means that the ellisoid is used as the reference.) Astrodetic and geodetic latitude differ at most by a few arc seconds ( .001 deg at most).
These complications arising from the shape of the earth don't stop there. For a spherical earth, the relation between the length of an arc on the surface and the angle the arc swings at the center of the earth is easy. It is just the radius of the intro times the angle in the natural units of radians. For the real world the same formula is used (history again), but the radius used is now both a function of position and direction.
Luckily all these effects are dependent on the amount the earth is bulged. On a percentage basis this is 1/3 %. As a rule of thumb, the maximum effects due to the figure of the earth in any of these areas is 1/3 % ( 1/300).
Another confusing factor is the angle units used in labeling coordinates. Locations are labeled with angles measured from inside the earth. These are usually expressed in degrees, minutes and seconds of arc. Some people are unfamiliar with these units.
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