21st June is mid-summer's day. As we go through the first half of the year, the sun gets slightly higher in the sky at noon (GMT) each day. It reaches it's highest point on 21st June, which is the 'summer solstice'.
So the ('official') middle of summer is defined by astronomy. However, the dates for the start and end of summer are only defined by convention. (So there can be differing answers depending upon which convention you follow). Many people (including, for example, the BBC's weather presenters) adhere to the tradition that the 1st of May represents the first day of summer.
For practical purposes, the convention followed by many owners' of holiday accommodation makes more sense. They often define 'summer' (for the purposes of determining their tariffs) as June, July and August (since those are the most 'summery' months in the UK). That makes 1st June the start of summer. As it's only a convention, this date is no more 'right' or 'wrong' than 1st May.
If we return to seeking a 'scientific' date, and accept that each season should last for a quarter of a year (i.e. around 91 days), it follows that the start and end of summer should be about 45 or 46 days either side of mid-summer's day. This would place the start of summer around 6th or 7th May.
Chris