So you have a number of questions here and I will try to cover them as best I can.

In terms of how you capture it and whether using your phone is OK or not that entirely depends on the board so if they do not specify and you are unsure it is best to ask in the specific game forum.

So livesplit is only as powerful as the tools the community have made for it. Some games have custom autosplitters written that can detect when to start/stop/split during a game but again you'd need to investigate for your specific game.

In terms of console it depends. I have seen some tools attempt to use image recognition for console games but I think for the most part the runners just hit the split button themselves at the correct point and for super competitive games might retime the run to get the accurate time.

Again with these specifics, the community for the game you are interested in will probably have a ton of advice to help you get started :)

Hope this can help a little.