Two way deterministic multihead counter automata or logspace TM with counter
A two-way deterministic (nondeterministic) multi-head finite automaton can be simulated by a logspace DTM (NTM), and vice versa.

So, for including class $ \mathsf{NL} $, you do not need a counter!

The value of the counter belonging to a two-way nondeterministic multi-head finite automaton with one-counter can be bounded by a polynomial, otherwise, the computation enters an infinite loop. Since both such a counter and the input heads can be simulated in logspace, $ \mathsf{LOGCFL} $ upper bound can be replaced with $ \mathsf{NL} $.

Note that any language recognized by a two-way deterministic (nondeterministic) one-head finite automaton with one-counter is in $ \mathsf{L} $ ($ \mathsf{NL} $).

There are many papers on these topics, e.g. this paper. You can find more after some googling. (Check also Lemma 1 of this paper.)