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Supergiant Fast X-ray Transients (SFXT) are High Mass X-ray Binaries displaying X-ray outbursts reaching peak luminosities of 10$^{38}$ erg/s and spend most of their life in more quiescent states with luminosities as low as 10$^{32}$-10$^{33}$ erg/s. The main goal of our comprehensive and uniform analysis of the SFXT Swift triggers is to provide tools to predict whether a transient which has no known X-ray counterpart may be an SFXT candidate. These tools can be exploited for the development of future missions exploring the variable X-ray sky through large FoV instruments. We examined all available data on outbursts of SFXTs that triggered the Swift/BAT collected between 2005-08-30 and 2014-12-31, in particular those for which broad-band data, including the Swift/XRT ones, are also available. We processed all BAT and XRT data uniformly with the Swift Burst Analyser to produce spectral evolution dependent flux light curves for each outburst. The BAT data allowed us to infer useful diagnostics to set SFXT triggers apart from the general GRB population, showing that SFXTs give rise uniquely to image triggers and are simultaneously very long, faint, and `soft' hard-X-ray transients. The BAT data alone can discriminate very well the SFXTs from other fast transients such as anomalous X-ray pulsars and soft gamma repeaters. However, to distinguish SFXTs from, for instance, accreting millisecond X-ray pulsars and jetted tidal disruption events, the XRT data collected around the time of the BAT triggers are decisive. The XRT observations of 35/52 SFXT BAT triggers show that in the soft X-ray energy band, SFXTs display a decay in flux from the peak of the outburst of at least 3 orders of magnitude within a day and rarely undergo large re-brightening episodes, favouring in most cases a rapid decay down to the quiescent level within 3-5 days (at most). [Abridged]