Answering as a mathematician myself, I think the problem is that math covers several different specialized subjects, and different specialties use the same words in different ways. For example the word "normal" is used in pretty much every specialty with a different meaning: it means a certain kind of subgroup in algebra, it means a certain existence property of real valued functions in set theoretic topology, and it means a space that essentially does not cross itself in algebraic geometry. In linear algebra it sometimes means "perpendicular" for vectors and sometimes "length one", and it means "commutes with its adjoint" for operators. The word "field", as a commutative ring in which all non zero elements have multiplicative inverses, is only used with that meaning in algebra. In differential geometry, or any area in which smooth manifolds are studied, a "field" is pretty much the same as the physicists think it is, i.e. a function defined on the manifold, with values which are either tangent or cotangent vectors, or tensors, or elements of some more general bundle, as weirdoguy said.
You will see this if you look in many textbooks on differential geometry or many geometric treatments of differential equations, e.g. Riemannian Geometry by Do Carmo, or Mathematical Methods of Classical Mechanics by Arnol'd, where the word "field" occurs in the index referring to the word in the physicist's sense, i.e. as a section of a certain bundle. Since the most commonly occurring bundle and the most important one is the tangent bundle, the word field in a differential geometry setting often means by default just (tangent) " vector field". My first mental response to the OP's question was therefore "vector field". In some books, such as Foundations of Differentiable Manifolds by Warner, or Notes on Differential Geometry by Hicks, you will find only "vector field" but not "field" in the index. In Lectures on Differential Geometry by Sternberg however, there is even a section on the "principal bundle" in which he describes how to define a field of "quantities" of any kind at all, as long as the fibers of the bundle in which the values are taken is any differentiable manifold having a certain linear group action.
Although this may make reading math seem confusing, actually all math texts try to be careful to state clearly what the words mean that they will use, just for this reason. I.e. in math we are not careful to always use words in the same way, but we do try to be careful to say in every setting just what the words we use will mean there. Of course we are also careless if we think the reader already knows what we mean. E.g. I don't think Arnol'd bothers to define a "vector field" in his somewhat more advanced book mentioned above, but he does define various types of them, "right invariant", centrally symmetric, etc... But when reading any math text, one should always try to verify what each key word means as used in that book. Hence a "theorem" in any given math book, means exactly a statement that is (hopefully) true as long as the words in that statement have the meanings that are given for them in that same book. This is why definitions are such a crucial part of math writing.
Sometimes you will find a math book in which a theorem you thought was difficult has been given a surprizingly easy proof. The trick is often that the meaning of the words in the theorem have been given in a different way from what you expected, precisely to render the proof easier, at the cost of perhaps proving a less precise statement. A notorious example of this to me is the "Riemann Roch" theorem, the most famous and perhaps most important theorem in algebraic geometry. Sheaf theoretic "proofs" of this theorem are often given so as to make the proof look absolutely trivial. Then at some point one notices that the definition of the "genus" of the complex curve has been given as the dimension of a certain cohomology group, and no attempt has been made to calculate that genus in terms of topology, as Riemann did himself. Thus the theorem obtained is not as strong as the original one. E.g. one may find a statement claiming only that the Riemann Roch theorem states that chi(D) - chi(O) = degree(D), whereas the full statement should include a computation of chi(O) in topological terms. In the 2 dimensional (complex) case, compare Th. 1.6 chapter V of Hartshorne, where the Riemann Roch theorem for complex surfaces is stated in the weaker form. A discussion of the additional information is mentioned in remark 1.6.1, and in the appendix A.4.1.2, a result known as Noether's formula, which implies it was due to Max Noether, hence was actually part of the earliest known form of RR for surfaces.
(Another confusion in this topic, which came up for me while writing the previous paragraph, is the use of the word "surface" to refer both to a complex curve ("Riemann surface"), which has real dimension 2 and complex dimension one, and also to a complex surface, which has real dimension 4 and complex dimension 2.)
