There is a ton of misinformation about this, however the answer is in your bible.
FAR/AIM 733 03/29/18
RESUME OWN NAVIGATION- Used by ATC to advise a pilot to resume his/her own navigational responsibility. It is issued after completion of a radar vector or when radar contact is lost while the aircraft is being radar vectored.
IMO, there is no use for this phrase in the IFR environment as you are already on an assigned altitude, and you're going to fly pretty much direct to your next destination. There are other phrases that better fit any IFR clearance or deviation.
This is really intended for VFR traffic temporarily in an IFR environment. For example, leaving controlled airspace, or loss of radar contact during flight following.
A while back there was an FAA Hottopic about this specific issue and the question was asked to 4 FAA inspectors what the phrase meant. All 4 unanimously stated that "Resume Own Navigation" allowed the pilot to determine their own appropriate VFR altitude.
Personally, I can say, in my 40+ years of flying, when I am flying VFR and hear that phrase, I go wherever I want at whatever alt I want, and I've never heard a peep from anyone. I can't recall ever hearing that phrase when flying IFR. If I did I think I'd be confused and I would have to question the controller about it.
There is a use for it in IFR. It was invented before Radar began being used for VFR services. It still exists. It did get changed from Resume Normal Navigation to Own Navigation back around 1980 or so. I guess someone didn't like the implication that Radar wasn't normal. It is used for VFR nowadays and is probably more so than the IFR Application now that RNAV is almost universal. Here are some of the Controllers rules on it:
e. Provide radar navigational guidance until the aircraft is:
1. Established within the airspace to be protected for the nonradar route to be flown, or
2. On a heading that will, within a reasonable distance, intercept the nonradar route to be flown, and
3. Informed of its position unless the aircraft is RNAV, FMS, or DME equipped and being vectored toward a VORTAC/TACAN or waypoint and within the service volume of the NAVAID.
PHRASEOLOGY−
(Position with respect to course/fix along route),
RESUME OWN NAVIGATION
c. Inform VFR participating aircraft when leaving the TRSA.
PHRASEOLOGY−
LEAVING THE (name) TRSA,
and as appropriate,
RESUME OWN NAVIGATION, REMAIN THIS FREQUENCY FOR TRAFFIC ADVISORIES, RADAR SERVICE TERMINATED, SQUAWK ONE TWO ZERO ZERO.
d. Inform VFR aircraft when leaving Class B airspace.
PHRASEOLOGY−
LEAVING (name) BRAVO AIRSPACE,
and as appropriate,
RESUME OWN NAVIGATION, REMAIN THIS FREQUENCY FOR TRAFFIC ADVISORIES, RADAR SERVICE TERMINATED, SQUAWK ONE TWO ZERO ZERO.
They don't cover 'resume own navigation' in Class C procedures. In all the Classes is this:
c. Aircraft assigned altitudes which are contrary to 14 CFR Section 91.159 must be advised to resume altitudes appropriate for the direction of flight when the altitude assignment is no longer required or when leaving Class B (or C or TRSA, my insertion, they all read the same) airspace.
PHRASEOLOGY−
RESUME APPROPRIATE VFR ALTITUDES
Resume Own Navigation is a horizontal function. That being said, If I had an altitude assignment and they said Radar Service Terminated, Squawk VFR, have a nice day, and left out RESUME APPROPRIATE VFR ALTITUDES, I would just say thank you and be on my merry way without questioning altitude. If they said Resume Own Navigation but didn't Terminate me and it seemed that it was time to get off that altitude assignment I would ask.