I'm on MacOS and iOS and haven't seen any of this. So I can only offer preferences.
- No savvy user will click on anything that says "install me," especially if it's browser related.
- I wouldn't even click a close or cancel button on such a popup, even if it had one.
- And if it didn't go away by clicking anywhere else on the web page, I'd close the window and be reluctant to ever open the page again.
- If it happened each time I visited the site (and I was newish to the site), I wouldn't come back.
- If the notification appears to be content on the web page itself, I'd be more likely to engage it. But floating in front of it smacks of a hack. If it's floating in front of the webpage (and moves when the window is moved), that's bad. If it's floating in front of the app, not part of the window, and looks to be generated by the OS, that's battle-stations-red-alert.
- I will never want a push notification. I never turn them on, or accept them, for anything except my own reminders.
- I get notified by email of Conversations activity, but not Alerts, and wouldn't want that functionality and settings to go away.
- I use TFP almost exclusively on Safari on Mac, and very rarely on any iOS device.
- When I want something to do, I go to the What's New page directly from a bookmark in Safari on Mac. I first look at the Alert and Conversations icons, then peruse the What's New list.
- If I ever got a TFP popup while working on some other webpage, or g-forbid while working in some other application than Safari, I'd go berserk and seek to turn it off, and then would be a bit put off that I had to waste time doing that. I would not immediately think "Neat, TFP is getting with the times," or anything of the sort. I would more likely think "I like things as they are, no need for 'improvements.'" I only want to opt-in to things, never opt-out.
- If there was a stand-alone TFP app for Mac, I wouldn't use it. And if that became the only option, I would likely be less active on the site. IMO, a forum belongs on a website.
I am probably not a typical user. I never type on a "smart" device if I can help. I don't like to read on a smart device either. I don't use social media apps, at all. I post comments here and there, but always while in a browser. I engage the internet world only when I feel like it, and wouldn't ever want to be bothered to do so when I was doing anything else.