H. Weyl:
starts from the vague notion: “symmetry = harmony of proportions” then gradually develops “first the geometric concept of symmetry in its several forms” and finally rises to the general idea, underlying all these special forms, namely that of “invariance of a configuartion of elements under a group of automorphic transformations.”
H. Weyl: Symmetry, Princeton: Princeton University Press, 1952, Preface, p. i.

Weyl made the generalisation here in two respects: to any transformation (operation) and to any kind of objects, but (cf., “a configuration of elements”) he restricted himself to geometric properties.
[N.B., this restriction concerns only his definition, and not all the usages of the term “symmetry” by him throughout his texts.]

”If I am not mistaken the word symmetry is used in our everyday language in two meanings. In the one sense symmetric means something like well-proportioned, well-balanced, and symmetry denotes that sort of concordance of several parts by which they integrate into a whole. Beauty is bound up with symmetry.
” H. Weyl: Symmetry, Princeton: Princeton University Press, 1952, p. 3. […]
(“… the second sense is which the word symmetry is used in modern times: bilateral symmetry” p. 4.

”Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection.”
H. Weyl: Symmetry, Princeton: Princeton University Press, 1952, p. 5.

R.P. Feynman's interpretation of Weyl's definition of symmetry:
“a thing is symmetrical if one can subject it to a certain operation and it appears exactly the same after the operation.”
R.P. Feynman, R.B. Leighton, M. Sands, The Feynman lectures on physics, Addison-Wesley, 1965, Vol. 1, Ch.11, p.1.

This interpretation – similar to Weyl – generalises the concept of symmetry in respect of the concerned object and operation, but not in respect of the property (cf.: "appears exactly the same").

J. Rosen:
"Symmetry is immunity to a possible change"
Rosen, J. Symmetry in Science: An Introduction to the General Theory. New York: Springer-Verlag, 1995. p. 2, p. 157

M. Petitjean:
"It is crucial to understand that mathematical constructs are models of real physical situations, and that a mathematical model of symmetry is a simplified image in our mind of some physical situation in which we would like to see symmetry."
This concept of symmetry gives rise to an unifying mathematical definition working in most pratical situations:
Petitjean, M. A Definition of Symmetry, Symmetry: Culture and Science 2007, 18[2-3], pp. 99-119.
(paper in open access: download PDF)