This question seems to stem from unfamiliarity either with how date strings are parsed by the built–in parser or with how they are formatted by various built–in toString methods.
The same parser is used by Date.parse and the Date constructor, it doesn't matter which is used in terms of how the string is parsed to a time value. The only difference between the two is that Date.parse returns the time value as a number, whereas the Date constructor uses the time value to create a Date instance (which is the only value that a Date instance holds).
The time value itself is milliseconds since 1970-01-01T00:00:00Z, so has no time zone (and means Date instances are effectively UTC). The value returned by getTimezoneOffset and the timezone displayed in toString methods (if there is one) is from the host system, it's not a property of Date instances or the Date constructor.
The timestamp pairs in your question represent exactly the same moment in time, but in different time zones.
Thu Jul 27 2017 13:10:42 GMT-0500 is Thu Jul 27 2017 18:10:42 GMT.
Thu Jul 27 2017 19:10:42 GMT+0100 (BST) is also Thu Jul 27 2017 18:10:42 GMT.
The value returned by Date.prototype.toString is entirely implementation dependent. When you send a date to output, the host may use any method it likes to represent the Date and may format it any way it wants. In the above, it seems the host timezone (UTC+0100) has been used with the default toString, which may return a differently formatted string in different implementations.
However, recent browsers seem to have settled on RFC 2822 format. Other methods that typically return different formats are toISOString and toLocaleString (noting that all three methods could return an ISO 8601 format string and be consistent with the ECMA-262, though I don't know of any implementations that do).
The next two dates also represent exactly the same moment in time:
Thu Jul 27 2017 13:10:42 GMT+0500 is Thu Jul 27 2017 08:10:42 GMT
Thu Jul 27 2017 09:10:42 GMT+0100 (BST) is also Thu Jul 27 2017 08:10:42 GMT
Again, the host timezone has been used to generate the output string.
As for "Firefox works in a similar way but it seems without the BST", see above. ECMA-262 only requires the the format be convenient and human readable. In this case, the browser developers have chosen to use an ISO 8601 format with no offset (i.e. UTC+0000).
So:
Thu Jul 27 2017 13:10:42 GMT-0500 is Thu Jul 27 2017 18:10:42 GMT+0000
2017-07-27T18:10:42.000Z is also Thu Jul 27 2017 18:10:42 GMT+0000
The choice not to use a timezone offset is entirely up to the browser developers and is consistent with ECMA-262.
If your issue is with how date strings are parsed, the duplicate is Why does Date.parse give incorrect results?. If it's with how dates are formatted, the duplicate is Where can I find documentation on formatting a date in JavaScript? The answers to those two questions should answer any issue raised in the OP.