What are different definitions of Universal Turing-machine?
I found some alternate definitions of Universal Turing Machine in papers related to the universality of small Turing machines and other models.

See for example the four definitions (the first 3 are equivalent and are similar to yours except that there there is also an explicit decoding function) in section 2 of Yurii Rogozhin, Small universal Turing machines, Theoretical Computer Science, Volume 168, Issue 2, 20 November 1996, Pages 215-240.

For example the Davis's definition in M. D. Davis, A Note on Universal Turing Machines ; Martin Davis, The Definition of Universal Turing Machine . Journal of Symbolic Logic 35 (1970), no. 4, 590; bypasses the encoding phase:

A generic machine (Turing machine, tag-system, ...) is universal if and only if its halting problem is r.e. complete.

Probably you're not interested in them (and you'll probably know them), but there are also  notions of semi-weakly universal  and weakly universal Turing machines (in which the initial tape has an infinite repeated string on the left/right/left+right of the input). For further details see the work of Neary and Woods on small TMs.