I've given this advice a lot but will try to summarize some main points.
ou have to be careful with indirect openers in daygame. She might think you are some salesman, religious person, or someone with an agenda
This is correct - specifically "crazy person" or "pan-handler" are some common ones women fear. When a girl doesn't know why you are approaching her, she has a lot of anxiety until she knows exactly why you are there. Most guys new to cold approaching have an ingrained (yet unwarranted) fear that girls will react negatively to a stranger coming up to them and saying, "You are stunning" that they try to beat around the bush which usually produces a worse outcome. MOST women are flattered when you are direct and it puts them at ease, even if they aren't interested in you, the interaction tends to go better.
My approach to day-game is always the same. My first choice is usually to go indirect but I will only do that if there is some situational thing to discuss. For example, if we are both waiting in a long line at a coffee shop, I might say something like, "I'm not one to wait in long lines but the coffee here is worth it. Have you been here before?" It has to be relevant to something you are both doing, experiencing, or looking for. Passing by a girl on a park bench who is reading a book - you wouldn't say, "Hey, sure is a hot day today." The "thing" you are talking about must be something somewhat unique or something that is impacting you both to an above-normal degree. Something you are "experiencing together." That's the goal anyway. One exception to this is if you are both standing and doing something very close to one another - like at the grocery store and she is looking at pears right next to you while you look at pears or apples - you can ask her a question about that. Notice that the key thing in these scenarios is that it should be fairly certain for her that you aren't a salesman or a panhandler.
If the above options aren't available because she's just walking down the street or sitting on a park bench, I find it best to be direct. When going direct, I almost always qualify it by saying "I know this is unusual but....." or something along those lines to let her know that you are socially calibrated and not a crazy person and you understand that approaching random strangers and stopping them to talk is not typical. This was more of a warm approach but one time I saw a girl staring at me as I walked by her in a store. My mind was elsewhere and I just wasn't in the mood in that moment to stop right there but after I got my things and checked out, I saw she was paying on another register as well. I walked outside the store and just stood there 15' from the door and waited for her. As she came out I just walked up to her and said, "Excuse me - I know it's a bit unusual to stop someone as they are leaving a store but you caught my eye in there as I passed you and I just wanted to meet you." She instantly smiled broadly and we had a great 10 minute long conversation. Lots of chemistry. Got her number and setup the date. I've also simply been on head-to-head collision courses with girls on the sidewalk and just stopped in front of them as we are about to pass and said, "Excuse me, I noticed how sexy you look in that jacket as you approached and knew I'd never forgive myself if I didn't take a moment to meet you," or something along those lines.
Talk for a few minutes depending on how the conversation is going. Make strong eye contact. Don't go a mile wide and an inch deep on their lives - go deep deep on just a couple topics. Get to WHY she is into the things she is, how it makes her FEEL. Then related it to something in your own life... doesn't have to be the same interest as her but it could be linked somehow, even if it's just linked to how it makes you feel compared to how hers makes her feel. After a bit say that you have to run but would love to continue the conversation sometime over a drink and tell her to give you her number so you can set it up.
