Improving Cadence on the Bike

What do you mean by "improve cadence"?  What's wrong with it?

The majority of evidence is that cadence should be self selected, and there is no ideal cadence, but most people will fall in the 80-100 range.

Ride at varying cadences and ride as often as you can.  However, cadence should rarely be your focus (nor should 'pedal stroke' or 'evening out the power').  Focus on riding harder/faster.  That's how you find the "right" cadence.  And it may change over time or at different intensity levels (up or down).

Being able to use a range of cadences is really the goal.  Yes, during short course races my average cadence is very close to 95, however I can ride anywhere from about 70-120 comfortably depending on the situation.  The way to "improve" it?  Ride using different cadences.

The litmus test would be being able to sustain power. what you described with "harder" fits better with force. They are not the same thing even though there is a relationship. As cadence goes up, force will drop if power is kept the same. Power is an energy rate. Your muscles are metabolizing energy to use. So what you're looking for is the best way for your body to put out this energy.

Yep.  You focus on generating power and making the bike move faster.  However you do that is "right".

You need a power meter. Without one, you are really just guessing and estimating the power you are producing. 

RPE and HR are both lagging indicators and indirect measurements of power, and also impacted by many other thigns.  Speed is a result, as is time on a course. The best way to measure power output is to measure it (surprise surprise).

Pedaling harder at lower cadence or lighter at higher cadence could be exactly the same wattage. Once you have a power meter, you can then tell if your FTP is higher if you ride at a higher cadence or higher if you ride at a lower cadence.

BTW, it's a myth that cycling at a higher cadence "saves the legs" for running. An often repeated, well cherished myth, but myth nonetheless. What usually happens is that people who ride at a higher cadence are producing lower power output than they would at a lower cadence, and this lower power output is what is "saving the legs".  An indirect outcome that could be equally replicated by simply pedaling softer without changing cadence.

TL;DR: it's power that matters, not cadence.

You need a power meter. Without one, you are really just guessing and estimating the power you are producing. 

RPE and HR are both lagging indicators and indirect measurements of power, and also impacted by many other thigns.  Speed is a result, as is time on a course. The best way to measure power output is to measure it (surprise surprise).

Pedaling harder at lower cadence or lighter at higher cadence could be exactly the same wattage. Once you have a power meter, you can then tell if your FTP is higher if you ride at a higher cadence or higher if you ride at a lower cadence.

BTW, it's a myth that cycling at a higher cadence "saves the legs" for running. An often repeated, well cherished myth, but myth nonetheless. What usually happens is that people who ride at a higher cadence are producing lower power output than they would at a lower cadence, and this lower power output is what is "saving the legs".  An indirect outcome that could be equally replicated by simply pedaling softer without changing cadence.

TL;DR: it's power that matters, not cadence.

There is more than one level of "hard" to ride at. The statement kind of encompasses them all together to help simplify the idea that yes, you do need to ride well above easy. Sometimes HR works well for these intensities, others it may not.