This a long comment rather than a complete answer.
Let me point out a paper of Bruce Blackadar
B. Blackadar, Shape theory for C* -algebras, Math. Scand. 56 (1985), 249-275.
where slightly more general conditions, which can be imposed in a natural manner on the generating relations, are considered. More specifically, in this setting the relations considered in the paper have the form
$$\|p(x_1, \ldots, x_n, x_1^*, \ldots, x_n^*)\|\leqslant \eta,$$
where $p$ is a polynomial of $2n$ non-commuting variables and $\eta\geqslant 0$. I am quite sure that this is not what you are looking for, though.
When the functions allowed in the generating relations are no longer polynomials but arbitrary Borel functions, it is difficult to talk about any kind of universality of such creatures. Indeed, in this case $h(f(a))$ need not be the same as $f(h(a))$ where $h$ is some *-homomorphism (these are not even well-defined a priori).
It is also possible to talk about C-algebras generated by order-zero c.p.c maps from matrix algebras $M_n$ etc (this is perhaps the order ingredient you have in mind). However those maps correspond precisely to ${}^*$-homomorphisms from $C([0,1], M_n)$ so this is the old notion of universality in disguise. An important example of a C-algebra which can be expressed in terms of (rather messy) relations involving order zero c.p.c maps is the Jiang-Su algebra $\mathcal{Z}$:
B. Jacelon and W. Winter, $\mathcal{Z}$ is universal, to appear in J. Noncommut. Geom., arXiv version.