symmetry of equilibria with heterogeneous players

the equilibria of the game in which the strategies of users who face the same reward and costs (i.e. same type) are the same.

This sounds like an ex ante (or ex interim or ex post) symmetric equilibrium, depending on the timing of the realization of the types. If both dimensions of the types are realized at the beginning of the game, then I'd go with the term "symmetric equilibrium", which is typically understood to mean the ex ante sort.

I think you may be confusing between "strategy" and "action", which may be why you rejected the term "symmetric equilibrium". In the context of Bayesian games, a strategy is a function of types while an action is a particular value of that function. For example, in a first-price sealed bid auction with uniformly distributed private values (types), the symmetric Bayesian Nash equilibrium involves each player bidding according to the same strategy function $s(v_i)=frac{n-1}{n}v_i$, where $n$ is the number of players and $v_i$ is player $i$'s value of the auctioned object, even though the actual bids will differ based on each player's realized type $v_i$. Nevertheless, players with the same value will place the same bid.