What is the proper way to clear contents?

A couple additions. Close isnt generally require Excel 2013+ (saving now recovers the used range). I generally reserve this for very large/problematic files in which closing and reopening may reduce its overall footprint in memory.

I usually Do like you have, Ctrl+End to get to the bottom right most cell in the used range. You then have to manually evaluate where your last used row AND last used column are. Its possible you have excess rows and/or columns (which will vary sheet to sheet). If I have both I generally start with the smaller of the 2 excess ranges (columns or rows). I click the first blank row/column's header and then Ctrl+Shift+Down arrow. That should select to the bottom or right most part of the sheet. I then check my status bar for any count/sum (if there is a count/sum then some of the cells selected are not actually empty and I go back to manually evaluating my true used range again). I will then try clear all first. If it works it generally sends you back to the first empty row/column selected (if not I later try delete). I then select a single cell at the top of the sheet (like A1) and hit save, then check the same range again. If it hadnt cleared I repeat the steps but instead after selecting the rows/columns I right click on their headers and select delete. This again should shift you to the first blank row/column. I then select a cell like A1 and save. Then recheck.

Clear all sometimes works, Delete seems to almost always work. So why use clear all? Well it can be faster and is less prone to triggering errors (like not enough memory). This is because you are clearing contents out of the cells, not removing them and replacing them with x number of new ones. In other words when you delete rows 500:1mil it removes them and then adds back a new set of rows from 500:1mil. Shifting cells/content in Excel can take alot of time, even if they are empty ranges.

There are some cases (especially at the end of a sheet, last row/last column) where deleting can cause ref errors in formulas and/or may not correctly clear out the range, and sometimes clear all works there.

Depending on what version of Excel you have, you may combine this manual process with something like enabling the Inquire addin (Excel 2013+) and using the clear excess cell formatting button (this clears things like fill colors in cells outside your used range).