Per the webpage you found, Egyptian citizens generally do need a visa to transit either landside or airside in London but can be exempted in some cases. Travelling to the US is one of those cases, having an EEA residence permit is another one.
You might be eligible for the ‘transit without visa concession’ if:
- you arrive and depart by air
- have a confirmed onward flight which leaves within 24 hours
- have the right documents for your destination (eg a visa for that country)
One of the following must also apply:
- you have a valid uniform format residence permit issued by an EEA state
The first three conditions must be fulfilled to qualify, which seems to be the case for you. Your Greek residence permit gives you the right to enter Italy as both countries are part of the Schengen area (I am assuming you have a regular Greek residence permit, if not then this answer might not apply).
But you only need to fulfill one of the other conditions in the list (in fact, it seems impossible to fulfill them all) and you told us that you do fulfill one of them (namely having a residence permit by an EEA state, as Greece is indeed a member of the EEA). So flying to Italy instead of the US should not be a problem if you fulfill the other relevant conditions.
That said, asking the embassy or even applying for a visa cannot hurt if you want to be absolutely sure (the visa does cost something). You can also ask your airline, as they will want to check your visa status before carrying you to the UK.