I am a Christian and do believe in the power of prayer, ie that prayer does change things and I have had many examples of that in my life. If I didn't I am not sure why I would bother praying. So, yes it is more than thinking about someone or holding them in your thoughts.
I am familiar with the type of situation you are talking about OP, and have been at meetings where we pray for people in that way.
I would never pray for someone in that context who had asked me not to. In fact I would think much less of the leadership of the church if, having been asked not to, they continued to pray for specific people in that way in a group prayer meeting.
I totally understand the feeling of violation, and that they are imposing this on you.
It is a breach of consent.
The real issue though is that they do this KNOWING you have asked them not to, and continue to tell you blatantly that they are doing this and going against your wishes. This is really a form of manipulation and control. They are violating your wishes, and it amazes me that they are so insensitive that they cannot see that.
There is a tendancy in evangelical churches to be very slack around issues like this, and things get shared 'for prayer' when they should have remained confidential. Eg, I was helping a lady from church, she had found a breast lump and was going to the emergency 2 week appointment. She asked for prayer, but I had not talked to her about what she wanted to share, so in the meeting I said - urgent hospital appointment, she is very anxious. People pressed for more details and I said - I am not sure what she wants me to share, so I won't say any more. Everyone was fine with that. But in many contexts it is all shared.
Another example is someone shared about an upcoming operation their ds was having and then said - he doesn't want everyone knowing but I told him this was his church family, so they needed to know so they could pray. At the time our dd was going through something and we had deliberately only shared it with one or two close friends as she did not want everyone to know.
That breach of their child's wishes actually made me very angry and I had several conversations with leaders around this issue of our teenagers right to confidentiality and setting an example. They got it.
If there is no respect, how can we expect people to want to join us anyway? The first thing I do is respect who someone is and what their opinion is, if we then have discussions about faith, fine, if they do not want to then I must respect that too. Otherwise there is a terrible arrogance abotu assuming I am somehow better than them, and that is not how God would have us be anyway.