Your plan 2 is the fast, direct, not especially scenic route with therefore no changes. Am I right that speed is more important to you than scenery on that route?
To see what I see go to http://www.sbb.ch/en/home.html which is after I have clicked on the "en" at the top for English.
Your cursor should be in the Timetable & Ticket Sales From box. Type Luzern. As you type it gives you several choices. Click Luzern (by itself) which indicates the Hauptbahnhof - main station. In the next box down type Lausanne, and again click on the city by itself. Don't enter a via. In the date box put the same day next week that you will be traveling next September because you want to see loadings on short notice, and in the time put 9:00 with the colon. Click on Search Connection.
Connection number 1 in the results will be the train we are talking about.
You can hover over any field and get more information. From left to right is connection result number, from/to, departure/arrival time, duration, number of changes, train type (travel with), Information, and Fare (from - based on half-fare card).
Under Information you will see 1.the three stick men, 2.the three stick men. 1 is for First Class, 2 is for 2nd class, and the three stick men are dark for there, white for not there. So for next week Tuesday, that train has 2 stick men in both classes.
Under the 1 on the far left, the connection result number 1, is a tiny little plus icon. Click that and then "show intermediate stops" and you can see all the platforms at all the stations that this train stops at, arrival and departure times for each intermediate station, and stick men for each station, seeing that it is medium occupancy all through its route until the last sagment when some people will have left the train.
Can you see all that?