Also a traffic engineer. That being said, the commentary below is general in nature, is not legal advice or an engineering work product, and is worth exactly what you're paying for it.
Most inductive loop detector types, if working properly and not damaged, can be adjusted to properly sense bicycles (or more precisely, metal bicycle rims) close to the loop wires without placing "false calls" from traffic in adjacent lanes. Diagonal-wire loops and quadrupole (sideways figure-8) loops are typically best for bike detection, while circular loops as used frequently in California don't have as good a reputation for detecting bikes. But this depends on whether the signal crew that maintains the signal is actually adjusting the loop sensitivities.
Phoenix Arizona has an online form at
https://www.phoenix.gov/streets/traffic-signal-issues for reporting signal problems such as detection failures. A decade or so ago, submitting one of these forms would typically result in a call a few days later from a signal supervisor saying the problem is fixed and to come out and test it. In the past several years, though, roughly coinciding with party-control turnover in the City Council (although elections are nominally nonpartisan), these complaints seem to be ignored, and when escalated get a "we don't adjust detectors for bicycles except on designated bike routes" response from the Street Transportation staff. And some of the signal detectors on designated bike routes such as 15th Avenue are remarkably bicycle-indifferent.
As for bicyclist behavior adaptations: Arizona does not have a specific exception for bicyclists at a signalized intersection which does not detect their presence.
ARS 28-645.C states that an an "inoperative" signal a driver may stop and proceed when safe, but this is generally intended for right of way regulation at dark signals due to power failure or other similar problem. To my knowledge there is no legal precedent that defines a non-detecting signal as "inoperative". What I've actually witnessed is bicyclists either running the signal (and vulnerable to citation under ARS 28-645.A.3), or executing a "right-U-right" maneuver, which may be legal depending on choice of location for the U turn, and allows the rider to choose when and where they make each part of that maneuver depending on number of lanes, presence of a median, and presence of other traffic. But it does involve extra time and distance.