Let us say I landed on runway 7 and will exit the runway thru taxiway Charlie, is that a standard report like "(Aircraft ID) taxing clear RW7 at Charlie"?
Nitpick: that would be runway 07, not runway 7
The phraseology is mostly standardised, and in general consists of notifying the relevant unit that you have vacated the runway. At a small, GA airfield with AFIS in the UK the typical communication goes as follows.
ATCO: G-ABCD Cleared to land Runway 07, wind 240 at 5
G-ABCD: Cleared to land, Runway 07. G-ABCD
... controlled crash ....
G-ABCD: Runway vacated at Charlie
This becomes a bit more formal at controlled airfields. In the UK, Cap 413 is the radio telephony communications bible. Secion 4.89 "Runway Vacating and Communicating after Landing" is the part you'e interested in. The communications would be more along the lines of (Note: CAP413 uses the name "Kennington" as a fictitious airport)
Kennington Tower: BIGJET 347, vacate left
BIGJET 347: Vacate left BIGJET 347
Kennington Tower: BIGJET 347, when vacated contact Ground 118.350
BIGJET 347: When vacated Ground 118.350, BIGJET 347
BIGJET 347: Kennington Ground, BIGJET 347, runway vacated
Kennington Ground: BIGJET 347, Kennington Ground, taxi to Stand 27 via taxiway Alpha
In North America, it's "clear".
"Vacated" is used in UK and possibly other places that use UK CAA practices as a basis of their conventions and rules.
You will almost never use the phrase at a controlled airport; you pull off the active at the end of the landing roll, and when tower sees you are pulling off onto the taxiway, just tells you to contact ground with the frequency.
At uncontrolled airports in North America you will always hear "XXX is clear the active".
The FAA has published an Appendix to the Pilots Handbook of Aeronautical Knowledge that gives this example of exiting the runway.
Initial Contact After Landing and Clearing the Runway
Pilot: Lincoln ground, November 123QY, clear of Runway two at Bravo, taxi to the ramp.
Controller: November 123QY, Lincoln ground, taxi to the ramp via Bravo.
You still hear some pilots say “Clear of the active.” but it’s mostly older pilots who were trained a long time ago and are set in their ways.
By saying who you are, what runway you just exited, and where you are, and (if it’s not obvious) where you want to go, ATC can more easily get you off of the movement areas and to your final destination.
It works at uncontrolled airports as well since landing traffic knows which runway you exited. That’s especially useful if there are multiple runways in use. It also helps taxiing aircraft because they where you are and where you are going so they know if they need to hold for you to use the taxiway or if they can proceed to the runway.