Should a clogged toilet where the clog is deep enough to cause sewage to back up into the adjacent tub be classified as a soft clog under any circumstances? I had such a clog and could not clear it myself. My landlord called out a plumber to fix and is now charging me $400 stating it was a "soft clog".

Just for your information.
A toilet drains into the main stack through what is called a toilet arm. A large bent pipe. Bath tubs drain into the main stack or into the toilet arm. When a blockage occurs after the point which the bath tub connects, water from the toilet will back up into the tub because the bottom of the tub is lower than the toilet bowl.

In general terms a "hard" clog or blockage is when the pipes are completely stopped up and no water will pass. A "soft" clog or blockage is when there is a blockage but water will slowly pass through over time.

In your case it is immaterial whether it was a hard blockage or a soft blockage. The main drain was blocked.

You should review your lease. It probably specifies that you are responsible for all blocked drainage pipes. Plumbers usually charge by how long it takes to clear the blockage. The further away and the more snake he has to put out the more he charges.

Remember,
"Never put anything down the toilet you haven't eaten first."