"xxxAir checking in at 360" I thought was a great idea - it let us know that he wasn't at our level. Equally "xxxAir passing 280 climbing 360". But when I did the same, my training captain said it wasn't necessary "Because ATC already know that!". Well, they might - but other aircraft won't and it all helps to build SA!"
The voice report used when switching freqs from one controller to another with altitude included, IIRC originated in the days when not all airspace was radar covered.
Even with radar coverage, if the controller you are checking in with does NOT copy your transponder, for whaterver reason, at his end or yours, alerting him to your altitude is a good thing.
I need to check voice reports again in the AIM to see what has chnaged since I used to teach this stuff. I agree with you on the wisdom of those reports.
However, "checking in at" is probably not the right report. (Again, I need to look this up).
Format from memory: "Houston Center, XXXAir NA556, Flight Level 250 (or one's altitude when FL is not appropriate)" is the standard call. If this has changed, I'd be curious as to why. Short, sweet, and to the point.
EDIT:
Looks like it is mostly the same, and I'd be interested to know if this is not the same in ICAO procedures.
From the 2012 edition of the AIM:
5-3-1.b.2. The following phraseology should be utilized by pilots for establishing contact with the designated facility:
(a) When operating in a radar environment:
On initial contact, the pilot should inform the controller of the aircraft’s assigned altitude preceded by the words “level,” or “climbing to,” or “descending to,” as appropriate; and the aircraft’spresent vacating altitude, if applicable.
EXAMPLE−
1. (Name) CENTER, (aircraft identification), LEVEL (altitude or flight level).
2. (Name) CENTER, (aircraft identification), LEAVING (exact altitude or flight level), CLIMBING TO OR DESCENDING TO (altitude of flight level).
NOTE− Exact altitude or flight level means to the nearest 100 foot increment. Exact altitude or flight level reports on initial contact provide ATC with information required prior to using Mode C altitude information for separation purposes.
(b) When operating in a nonradar environment:
(1) On initial contact, the pilot should inform the controller of the aircraft’s present position, altitude and time estimate for the next reporting point.
EXAMPLE−
(Name) CENTER, (aircraft identification), (position), (altitude), ESTIMATING (reporting point) AT (time).
(2) After initial contact, when a position report will be made, the pilot should give the controller a complete position report.
EXAMPLE− (Name) CENTER, (aircraft identification), (position), (time), (altitude), (type of flight plan), (ETA and name of next reporting point), (the name of the next succeedingreporting point), AND (remarks).