Boyd’s page for this Woodward family has Elizabeth born 1583, with 2 suggested marriages for her, both queried:
(1) George Cater, 1602
(2) Thomas Style, no date.
There is no mention on the page of Anthony Smith.
The parish registers show that an Elizabeth Woodward married George Cater at St Dionis Backchurch on 15 June 1602.
Also, an Elizabeth Cater married Thomas Styles on 30 Oct 1604 at St Leonard Shoreditch (no mention of her marital status).
Boyd also cites a PCC will of 1601 for Elizabeth’s father John Woodward, citizen and ironmonger, who died 13 Nov 1601, buried at St Dionis Backchurch. This will can be seen on Ancestry.