I need to remind you that I have no electrical appliances, connectors or anything like that. A tape and a pliers are probable the only things I know how to use. My plan now is to tape around the green wire and repeat the same process for all other wires.
"When all you have is a hammer, everything looks like a nail." Your procedure will work. It won't look "pretty" or "professional".
... Is this kind of connection safe? Also, is there a huge between my method and others (with soldering...)?
Safe is a relative term. If firmly pulled, the wires could slip out of your pliers-squeezed connection and the bare ends short against something. Soldering would help to prevent that kind of "accident" from occurring.
The sharp edges of the existing pin receptacles after flattening with pliers (one of which you have already ruined by "crimping") need to be thickly wrapped to prevent "break through" of the plastic tape insulation, which will creep after wrapping. IMO, you would have been better off if you had just cut them off and spliced the wires together with a soldered connection and then taped the spliced connections. In heavy electrical wiring (which apparently this is NOT), electrician's plastic tape (or self-vulcanizing rubber tape) is traditionally over-wrapped with "
friction tape" to provide increased mechanical protection of the spliced joint.
... I am asking because I know an electrician who could do it, but if there is no significant difference, then I would stick to my method. Also bear in mind that this device would be used in a professional company so I would like it to look as best as possible. Would other electricians think that I am a complete idiot or they wouldn't care?
Why would they care? It isn't their problem. If you wanted it to look as best as possible, you should have hired a professional electrician to make the connections. Then it would become their responsibility to make it safe, pretty, and professional looking.
Your method will probably work. If it doesn't come apart, or overheat and melt the plastic tape because of excessive resistance in the "pliers-crimmped" connection, or cause arcs and sparks, a heavy overall wrapping of tape will hide the details. You do what you have to do, given limited tools and available parts to work with. A "correct" solution would have been to insert the original pin-receptacles in a mating connector and crimp mating male pins to the other four wires and insert them in another mating connector. Since pins and connectors are not available, you have to improvise. So, good luck with that.