Following today's update (and as announced before in this thread), this text should answer your questions 
About "No time limit" and "Training mode"
If you play a game without time limit, you cannot expel players. It also mean that you are "trapped" in the game if the others players cannot play. This is why "no time limit" should be only played in game with "no penalties" and "no ELO": this way, you can leave the game anytime without any damage to your reputation or rankings. And this is exactly what "training mode" provides: a safe way to play with "no time limit".
We let the possibility, in "old" lobby, to play without time limit and without the ELO, and it was probably a mistake for the reason explained just before. This possibility has been removed today.
If you want to want to play with "no time limit", we now encourage you to play using "Training mode" ("infinite sign" in the new lobby), which is the safest way to play this way.
If you really want to play have a ELO gain/loss after your game, we now encourage you to create a Turn based game with the slowest speed (1 move every 2 days). If you were used to use "Normal mode + no time limit", it makes practically no difference: even if some of your opponents goes over their time limit (which is pretty high), you are absolutely not forced to expel them. However, the improvement compared to the previous situation is that you still have the possibility to expel a non-playing player in case it is needed.
About "No time limit" and "Training mode"
If you play a game without time limit, you cannot expel players. It also mean that you are "trapped" in the game if the others players cannot play. This is why "no time limit" should be only played in game with "no penalties" and "no ELO": this way, you can leave the game anytime without any damage to your reputation or rankings. And this is exactly what "training mode" provides: a safe way to play with "no time limit".
We let the possibility, in "old" lobby, to play without time limit and without the ELO, and it was probably a mistake for the reason explained just before. This possibility has been removed today.
If you want to want to play with "no time limit", we now encourage you to play using "Training mode" ("infinite sign" in the new lobby), which is the safest way to play this way.
If you really want to play have a ELO gain/loss after your game, we now encourage you to create a Turn based game with the slowest speed (1 move every 2 days). If you were used to use "Normal mode + no time limit", it makes practically no difference: even if some of your opponents goes over their time limit (which is pretty high), you are absolutely not forced to expel them. However, the improvement compared to the previous situation is that you still have the possibility to expel a non-playing player in case it is needed.