Does an SVFR clearance extend to Echo surface extensions to Class D airspace? No it does not.
Likewise at KTLV (Lake Tahoe), authorization to operate under special VFR would only be valid within the round dashed magenta shape, not the rectangular dashed magenta shape. (See chart).
In both cases the key phrase is "designated... for an airport".
Therefore the only practical purpose of a Class-E-to-surface "extension" is to extend standard Class E cloud clearance and visibility requirements all the way to the surface, for the protection of IFR traffic emerging from clouds at low altitudes. Once a VFR aircraft is above the level of the floor of the Class E airspace surrounding a Class-E-to-surface "extension" (typically 700' AGL), there is no practical significance as to whether the aircraft is within the "lateral boundaries" of the "extension" or not.
Highly related -- January 10 2018 FAA memorandum carefully addressing the precise meaning of "within the lateral boundaries of the surface area of Class E airspace designated for an airport"-- the specific context here is commercial drone operations. This is virtually the same language as we find in FAR 91.157 -- "within the airspace contained by the upward extension of the lateral boundaries of the controlled airspace designated to the surface for an airport", and FAR 91.155 -- "beneath the ceiling... within the lateral boundaries of controlled airspace designated to the surface for an airport when the ceiling is less than 1,000 feet."
Also related -- much longer in-depth answers to these questions--
Does FAR 91.155c apply to class E surface extensions?
Which parts of class E airspace can an ultralight (part 103) fly in without prior ATC authorization?
As an aside, in relation to interpreting the language of the ATC Handbook regarding where SVFR is authorized (7110.65W, para. 7-5-1-a 2), which lacks the "designated for an airport" or "designated to the surface for an airport" qualifier, it is relevant to note that the AIM, and the section headings of FAA JO 7400 "Airspace Designations and Reporting Points", both use the phrase "Surface Area" to refer ONLY to the Class-E-to-surface airspaces that actually enclose the airports whose approaches they protect-- i.e. the same class-E-to-surface airspaces that are described as "designated as a surface area for an airport" in the main body of JO 7400-- and not to the Class-E-to-surface "extensions". The "extensions" are never called "surface areas" in these documents.
If anyone can come up with ANY written guidance from any FAA source on the exact original question (or the one re FAR 91.155c), that is written SPECIFICALLY to address SVFR operations, and gives an EXPLICIT clear answer without making the reader parse the regs, AIM, and other relevant materials to come up with an understanding of the meaning of phrases like "designated to the surface for an airport", "within the lateral boundaries of controlled airspace designated to the surface for an airport", "within the lateral boundaries of the surface area of Class E airspace designated for an airport", "Designated as a Surface Area", and "Surface Area", then PLEASE add a link to that material here. Regardless of whether the material is recent or older, or supports or contradicts this answer. Likewise for that matter for any content SPECIFICALLY addressing the ultralight question re FAR 103.17. Research on related topics suggests that it would not be at all surprising if the FAA has not been entirely consistent with its memoranda, rulings, and interpretations on these issues.