There are a lot of definitions of 'manipulate' etc around, using slightly different wording and emphasis. Here are a few:-

Manipulate: To control or play upon by artful, unfair, or insidious means especially to one's own advantage. (Merriam Webster)

Manipulate: To influence or manage shrewdly or deviously: (FreeDictionary)

Manipulation: Attempts to control or play upon others' hopes or fears to attain selfish ends while disregarding their aspirations or well being. (Business Dictionary)

Manipulative: Skilful at influencing somebody or forcing somebody to do what you want, often in an unfair way. (Oxford Advanced Learners Dictionary)

A recurring theme of the definitions is that the manipulator's objective is their own personal advantage. LR does not benefit directly from Anne's decision to end the engagement, and she is acting in what she honestly feels is Anne's best interest. However, she is imposing her personal prejudices on Anne:

LR "..had prejudices on the side of ancestry: she had a value for rank and consequence," (Ch.2)

(Sir Walter) "... thought it a very degrading alliance; and Lady Russell, though with more tempered and pardonable pride, received it as a most unfortunate one.
Anne Elliot, with all her claims of birth, beauty, and mind, to throw herself away at nineteen ......... a throwing away, which she grieved to think of! Anne Elliot, so young; known to so few, to be snatched off by a stranger without alliance or fortune" (Ch.4)

I think it is clear that LR's prejudice in favour of rank and consequence is partly driving her disapproval of the match with Captain W. (We know that Anne does not share this view - in Ch.6 Henrietta says that "all the world knows how easy and indifferent you (Anne) you are about it".) Then there is this:-

"His sanguine temper, and fearlessness of mind, operated very differently on her. She saw in it but an aggravation of the evil. It only added a dangerous character to himself. He was brilliant, he was headstrong. Lady Russell had little taste for wit, and of any thing approaching to imprudence a horror. She deprecated the connexion in every light." (4)

This is all about LR's personal dislike of CW. As it happens, the character traits she dislikes are the very ones which make him successful in his career and, I suspect, traits which are very appealing to Anne, but that's by the by. Surely what matters here is whether Anne likes him. LR's likes and dislikes are not grounds for Anne to give him up.

Another recurring element is the sense of influencing someone in a devious, insidious or unfair way. I have no doubt that LR stated some reasons for her opposition very plainly, but I cannot imagine that she was explicit about how her personal dislike of CW influenced her point of view. Is that not devious?

LR seems to ignore Anne's attitudes and ambitions except when it suits her purpose:
"The belief of being prudent, and self-denying principally for his advantage, was (Anne's) chief consolation".

Here LR is taking unfair advantage of the fact that Anne wants the best for CW to persuade her to give him up. That is manipulative, IMO. Anne is in no position to judge what is in CW's advantage, and LR has no right to do so. The only person who can make that judgement is CW himself.

One final quote:
"She was persuaded to believe the engagement a wrong thing -- indiscreet, improper, hardly capable of success, and not deserving it."

We are not told how LR persuaded Anne to this extreme and irrational conclusion, but it cannot have been by reasonable arguments. LR may not have been absolutely manipulative here, but I do think she was unfair and was taking advantage of Anne's trust and reliance on her to attain her own ends.

I think that LR was quite right to be concerned about the imprudence of Anne and CW marrying before he had progressed in his career and made enough money, and I have no problem with her persuasion of Anne on this account, using reasoned argument. But think she was manipulative to impose her own personal preferences and prejudices on Anne and to ignore Anne's views. She was manipulative when she convinced Anne that she would be acting for the benefit of CW, and she was manipulative when she decided "It must nor be", and caused Anne to end the engagement rather than simply wait until CW had enough money to support a wife and children. LR wanted that relation to end no matter what, and IMO the tactics she used to achieve that were definitely manipulative.