Despite the endless Internet memes and folklore, the ITU is the standards body that defines what 2G, 3G and 4G all means.
The controversy and surviving misinformation began in 2010 and the Sprint HTC EVO 4G - the world's first 4G phone, based on WiMAX.
Released before the 4G spec finalized - but designed and labeled back when it was clear that WiMAX would be the new 4G.
Except that the committee pulled the spec and sent it back for study - as they had repeatedly over the last year and more.
The other carriers countered that they were as fast as WiMAX and knowing that the public didn't know better, all claimed to be 4G.
Including LTE, which was no more 4G than WiMAX or HSPA+.
Within months the situation moved past intolerable when the ITU released the 4G spec that said -
To be 4G, you had to be on either WiMAX 2 or LTE-A and provide 100 Mbps downloads.
That was in 2010.
The industry revolted and in December 2010, the ITU capitulated on the definition - but not the actual spec -
http://www.phonearena.com/news/ITU-says-LTE-WiMax-and-HSPA--are-now-officially-4G_id15435
So, LTE officially became 4G in December 2010, right alongside HSPA+.
As to which is faster or better depends on your service area and network load vs its capacity. Lots of people get better HSPA+ performance than LTE users.
And until you start getting +100 Mbps service, it's kind of pointless to worry about what the real 4G is.
All of the current 4G services offer performance that exceeds the broadband rates most people had at home just a few years ago.
The 4G speed debate comes down to what you really need on a phone.
The max AV bit rate for a Blu-ray movie is 48 Mbps.
Then again, more people watch HD at 6 to 8 Mbps with Netflix that like most modern services, uses modern adaptive technology.
The rest of the question about the difference between the two, HSPA+ and LTE, has been answered - so now you know what 4G means.
HSPA+ is as "genuine" 4G as LTE.