Certification, then evaluation
Operators must be certified by an accredited certifying body, NCCCO being the best known, by type and, where the certification scheme distinguishes it, capacity.
Since the 2018 final rule, certification alone is not enough. The employer must also evaluate the operator as competent to operate the specific equipment safely, and document that evaluation. Certification says they can run that class of crane; evaluation says they can run yours, on your work.
Riggers and signal persons are separate roles
Anyone rigging a load must be qualified for that work, and signal persons must be qualified too, with their qualification documented. On a bare rental you are supplying all of these people.
Bare rental moves the whole burden to you
Bare rental means you supply the certified operator, the riggers, the signal person, the lift plan and the insurance. Operated and maintained means the crane house supplies the operator and usually an oiler, and carries the operational responsibility with them.
In Texas that choice is also a tax difference: equipment supplied with an operator and billed as a single charge is generally not treated as a taxable rental of tangible personal property at all.
Ground conditions are the crane user's duty
The controlling entity must ensure ground conditions are adequate to support the crane. Outrigger pads, crane mats and a check for voids or backfill under the setup area are not optional extras. On Gulf Coast fill in particular, they are the lift.