The modern LED that uses transparent substrates with back reflectors and AlGaAs AlInP type materials are classified as HB types or high brightness.
"High Intensity" is simply a marketting term.
You can get light out of any LED with a few mA and get much more light with a power module than a 5mm LED with 3mA but that is not really cost effective. Power Modules tend to have wide beam angle and for "Indicators" vs Luminators" the low current LEDs have narrow beamwidths as low as 6 deg. for more "brightness" when viewed on center axis although power level is very low. ( < 75mW)
Physical classifications more often used are; Discrete, HB and "Power modules"
Packaging styles vary by the hundreds.
Current ranges vary from 10mA to 10Amp for arrays.
Lumen power depends on input power and substrate efficacy, while Brightness can be modified with reflector lens (cups) built-in which can increase the Luminous Intensity , Iv, "almost" double by reducing the beam angle by 50% until lens losses overcome the gain.
"Low Current" I presume means these are indicators and generally rated at the "de facto" 20mA standard, which by the way is limited by the heat dissipation from epoxy encapsolation which protects it but unfortunately is a thermal insulator.
Higher Current discrete parts have an improved lead frame for better dissipation of heat and can handle 50~100mA.
Power Modules are built on Aluminum substrates for better heat spreading and are rated for 0.5W, 1W 3W 5W,10W 50W 100W etc. and may be single junctions or arrays of junctions. Since voltage varies with internal junction temperature, they are generated tested on a cold sink @25'C at constant current.
You can achieve > 200 Lm/Watt at low currents today, but at a high cost/Lumen factor so demand is to increase Lumens/$ at a standard CRI , colour temp and rated power level.
There are many factors which increase Lm/Watt in your choices of existing production LEDs;
example of Sensitivity factor;
S(ζ/If)= -20% ... efficacy,ζ, increases 2% for a 10% drop in If ( & visa versa)
Here are some experience based Sensitivity factors.
The above graph shows the range of Lumens for different part numbers at a fixed colour temperature for the OSRAM LUW CQAR LED.. This part is intended for "streetwhite" colour and not simply warm or cool. 5500'K is considered daylight avearge white. The reason efficacy improves with cool LED's is simply there is less phosphor coating and conversion loss.
Other negative factors for efficacy are;
- use poor heatsink thermal resistance and operate much higher than rated 25'C test
- exceeding -5V from E fields or staic
- excess soldering temps
- over-drive the current with PWM peaks to achieve constant average intensity
- mfg choice of wafers, reflectors, lens, coatings and manufacturing processes are all significant variables beyond the scope of this answer,.