The "best" solution would be to use brand new genuine pinball flipper and bumper coil assemblies. But then, of course, you would be working with dangerously high voltages, 40V+ for some coils. Somebody clever (Chriz99?) hit on the idea of using used Siemens industrial contactors as a cheap, safer and sufficient simulator of the genuine pinball part. And the rest of us for the most part have followed along because it is at least a known quantity. In the process, we drive up the prices of used contactors on one another.
A lot of people used the Siemens model number you referenced. Another model, what I have bought is 3RT1016-1BB41.
One tip is that you might search for something like 3RT1016-2BB41. Identical to 3RT1016-1BB41 in every way except that it has crimp terminals instead of screw terminals.
There are tables from other manufacturers such as Allen-Bradley that allow you to cross-reference their contactors with Siemens so that you can identify a part that is comparable to a Siemens model number. Google "allen-bradley siemens contactor cross reference"
I wouldn't even say that all contactors have to be the same manufacturer and model. If you are buying used then the condition of these contactors will vary and so will their output.
Usually, parts from other manufacturers are cheaper than Siemens, but I haven't seen much in the forums that would conclude that they would not perform as well as Siemens. Or that you would not be just as happy with them as you would be had you spent extra for Siemens contactors. There's a few but not many head-to-head comparisons in the forums on contactors. In scouring the forums over the past few months on this subject, I've not seen the research that compares contactors to real parts.
You have genuine parts already. If you can handle the electrical, I'd suggest giving it a try.