You might want to have a look at Section 3 of Chapter 2 of the Introduction of Saul Lubkin's Cohomology of Completions (North-Holland Mathematics Studies 42, 1980), where the differences between the two possibilities of defining a category of graded vector spaces suggested by you (with the obvious correction suggested by Mark) are discussed in detail and in a very general setting.

You might want to have a look at Section 3 of Chapter 2 of the Introduction of Saul Lubkin's Cohomology of Completions (North-Holland Mathematics Studies 42, 1980), where the differences between the two possibilities of defining a category of graded vector spaces suggested by you (with the obvious correction suggested by Mark) are discussed in detail and in a very general setting.

Also related, less abstract, but still very interesting are the following two papers (for whose authors names I am not capable of producing here the necessary diacritics):

J. L. Gomez-Pardo, C. Nastasescu, Topological aspects of graded rings, Comm. Algebra 21 (1993), 4481-4493;

J. L. Gomez-Pardo, G. Militaru, C. Nastasescu, When is HOM$_R(M,-)$ equal to Hom$_R(M,-)$ in the category $R-gr$? Comm. Algebra 22 (1994), 3171-3181.