There is unlikely to be any efficient algorithm.
Your first class of constraints are monotone exactly-1 CNF clauses. Your second class of constraints are monotone CNF clauses. The monotone part indicates that negated literals aren't allowed (you can't have $x_1 - x_3 = 1$ or $x_1 - x_4 \ge 1$).
Thus, in the special case where you have only type-2 constraints, the problem becomes #monotone-SAT. Unfortunately, this problem is known to be hard. #SAT is #P-complete, and monotone #SAT is #P-complete as well: it is #P-complete even for monotone 2CNF clauses (i.e., type-2 constraints with only two variables). It is also known that it is NP-hard to approximate the number of solutions. As a result, there is unlikely to be any efficient solution unless the number of variables and constraints is fairly small. Of course, your problem (with a mixture of type-1 and type-2 constraints) is potentially even harder.
So what can you do, to make the best of the situation?
One approach is to code this as an instance of #SAT, and try applying some off-the-shelf #SAT solver. You can encode type-1 constraints in SAT using the methods described at Encoding 1-out-of-n constraint for SAT solvers.
Or, you could express the constraints as a BDD and then apply model-counting methods for BDDs. I expect this to perform worse than a #SAT solver, but you could try it.
Another approach is to use an approximation algorithm. There are existing algorithms for approximate-#SAT, though they too will hit a limit if you have too many variables and/or clauses.