There is another definition, used in works like Thompson's (1917) On Growth and Form and Alexander's (1964) Notes on the Synthesis of Form.
For Alexander, "physical order," "organization," and "pattern" are all approximate synonyms to "form." As in rschwieb's answer, the form is a concrete realisation of something -- a simple name for this something is "function."
However, in general "function" is too specific and telelogical; another broader term in Alexander's lexicon that also describes what gives rise to form is "forces."
Some added complexity along these lines comes from, e.g., Turing, whose paper on "The Chemical Basis of Morphogenesis" is basically a continuation of Thompson's line of thinking. Morphogenesis is basically a shorthand for "the growth of form." The philosopher Simondon also thought about information this way, and Sloman describes how this philosophical usage relates to colloquial understandings of the term.
What all of this has to do with the formalism of, e.g., Hilbert is quite interesting. In this sort of mathematical formalism, what is considered is manipulation of statements according to certain given axioms and inference rules. This is similar to the usage of Jazwinksi in the question. Another example would be "formal manipulation of Taylor series," famously used by Euler. Here, certain rules are followed algorithmically, without concern for whether they are entirely legal, or for what the results "mean." See André Nicolas's comment about "symbol-manipulation" -- but keep in mind that to be "formal" there do need to be some underlying rules governing the symbol manipulations, they are not arbitrary.
Contrary to the popular mis-attributed quote, Hilbert did not think of mathematics as a "a game played according to certain simple rules with meaningless marks on paper." Russell did seem to think about it this way, since he said:
Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.
Even if this is true (!) it is a rather extreme view. For Hilbert, rather than truth, the relevant consideration would be plausibility:
In discussing the method of mathematics, I have already stressed that when building a particular theory, it is a fully justified procedure to assume still unproved, but plausible, theorems (as axioms), provided one is clear about the incomplete character of this way of laying the foundations of the theory.
To simplify somewhat, a certain kind of elementary "formal" argument would be one along the lines of "If it looks like a duck, swims like a duck, and quacks like a duck, then it probably is a duck." This may be surprising, because it makes reference to observations, not just axioms or definitions (although it does refer to those). What's interesting though is that the observations explicitly refer to observed form (in the Alexander/Thompson sense described above). In this restricted setting we see, again, that the symbol manipulation is not arbitrary, and that it is guided not only by rules, but also by observations. In the same way, formal manipulation of Taylor series is only used when it is useful for the purpose at hand!