I would agree that there is a very thin line between anti-Semitism and anti-Zionism and there is in the minds of some confusion in regard to differentiating between the two. However, to state that the foregoing is the sole of the problem within the Labour party would be wrong, as there are "a number" in the party who without a doubt are anti-semitic.
In the above, Jenny Formby "laid the problem firmly on the line" to Labours National Executive Committee in stating that the party has a membership of over half a million and is the largest political organisation in Europe. The Labour movement in the Country has a total Membership of six million through the trades unions, with many individual union branches using their funds to affiliate to district and constituency Labour parties. In that way, those trade union members also speak out with their views at meetings etc as affiliated members.
Therefore, what Formby started to the NEC so forcibly, was that it is impossible to police what every member in the movement states, and that is just one of the problems in being such a large organisation.
The above I feel states the problem of anti-Semitism in the Labour Party as it really is. The party also has "a residual" Blairite faction within the parliamentary party that have never accepted the changes in the movement since 2015 and have used every opportunity to undermine the leadership with these allegations. There is also the extream right-wing press in Britain that have worked in conjunction with those disgruntled MPs to maximise the impact of anti-Semitism in the public mind.
However the size of the party is the major issue in this matter, and that can be judged against the Conservative party were with a total membership of only one hundred thousand, they have not been able to eliminate Islamophobia in that organisation.