Nobody should play any prescribed "role". That is acting according to a script. It is not a good way to start a relationship. For a relationship to be healthy and productive everybody is going to have to be able to improvise, compromise, adjust, etc., etc. about, oh, 99.999999999999999999999999% of the time.
Scripts are for theater and film.
Nobody should deliberately formally "initiate". People make it sound like either the man or the woman has to consciously decide to formally "ask" the other "out" and find the courage to do it. The way that it probably usually happens is that in the course of normal spontaneous conversation somebody says something like, "Why don't we continue this conversation at..." Nothing planned. Nothing deliberate. Certainly nothing requiring any courage. Just letting a relationship take its normal course.
How does a person show interest in another person? You talk to them. You ask them questions about themselves. You show them that you are paying attention ("I saw you at...the other day"; "I saw that your research project is being considered for..."; etc.). Nothing that you don't do already with all of the other people in your life. Nothing complicated.
Some people, due to variations in temperaments, communication styles, etc., may be more bold, aggressive, direct, etc. as they manage a relationship (let the relationship take its natural course) and show interest. To what degree one should be his/her usual self or adjust (be less direct, less aggressive, etc.) probably mostly depends on what the other person is comfortable with. That is probably what is meant by "coming on too strong".
In other words, no right or wrong way of showing interest applies to every member of a gender/sex. To know what works one is going to have to see how the particular individual that he/she is interested in responds.
Add it all up and you will see that it takes a lot of work by both parties.
Being deliberate and trying to follow a script with a formal initiation, "roles", etc. compounds that work. More work--unnecessary work--means that much less time and energy can be spent enjoying the relationship.
Deliberately "pursuing" a person is objectifying him/her.
If you want to relax and with minimal anxiety build a relationship based on mutual respect then let the relationship develop spontaneously.