When a localizer is offset at an angle with the runway track, it functions as a "Localizer Directional Aid or LDA." What's the exact offset degree dividing them? Any offset degree greater than 0 makes a localizer an LDA? I've checked various sources including AIM, but none of them gives me a definite answer.
The maximum angular offset for a LOC is 3° for FAA and 5° for ICAO. Everything else is named either LDA (FAA) or IGS (ICAO).
In ICAO wording, any ILS which doesn't meet all ILS requirements is an instrument guidance system (IGS).
Localizer type directional aid (LDA) is a designation used by FAA.
Guidance /aid refers to the fact that this system is not a precision approach, only an assistance for for the approach.
The most infamous IGS (at least for sim users) is probably the one used at Kai Tak RWY 13 until the airport was decommissioned. Current ones include Sion LSGS RWY 25 which terminates before the runway (see also Washington KDCA RWY 19 as commented by @Gerry), Innsbruck LOWI with a 5° offset localizer or Lugano LSGS RWY 01 for its 6.6° glideslope.
Many of those non-standard approaches are required to access an aerodrome at the bottom of a valley. All require special training and certification before a pilot is authorized to use them.
ICAO defines approach characteristics in Doc 8168 Procedures for Air Navigation Services (PANS-OPS), volume II: Construction of Visual and Instrument Flight Procedures. According to SKYbrary (Eurocontrol) "PANS-OPS procedures are the international standard and are used throughout Europe and in many other countries world-wide".
FAA equivalent are the United States Standard for Terminal Instrument Procedures (TERPS, Order 8260.3). According to the same source: TERPS are used by "USA and certain other countries. These include Canada, Korea, Saudi Arabia and Taiwan".
Final approach segment:
(5.2.1) When runway aligned track guidance is not possible it may be offset up to 5 degrees without OCA/H penalty (see 188.8.131.52, “Aligned straight-in approach”). Beyond these limits (or where other requirements cannot be met) a circling approach shall be used.
ILS Cat I:
(2.1.1) The localizer course line shall intersect the runway extended centre line:
a) at an angle not exceeding 5°; and b) at a point where the nominal glide path reaches a height of at least 55 m (180 ft) above threshold. This is called intercept height.
(2.1.2) The procedure shall be annotated: “localizer offset ... degrees” (tenth of degrees). The general arrangement is shown in Figure II-1-2-1.
Localizer without glide path. Final Segment alignment (8-1-2):
Localizers aligned within three degrees of the RCL are identified as localizers. If the alignment exceeds 3 degrees, they will be identified as LDA facilities.
Localizer with glide path (ILS). Final segment alignment (10-2-2 a):
Where a unique operational requirement indicates a need to offset the course from RCL, the offset must not exceed three degrees. The offset course must intersect the runway centerline at a point 1100 to 1200 feet inside the DA point (see figure 10-2-2 ). For offset courses the minimum HAT is 250 feet and the minimum RVR is 2400.
For LDA with GS (10-2-2 b):
The final course maximum offset from RCL extended is 15 degrees. The final course must cross the RCL extended at least 3000 feet from LTP, but no more than 5200 feet from LTP.