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That's a really interesting question and says a lot, even in the asking. Questions about the genuineness of our faith and the security of our relationship with Jesus seem to be never far away for many people. Mostly, it's a sign of an insecure but genuine faith and is something that can be worked-on over time. Maybe the best way of working on it is not to focus on it but, instead, to fix our eyes on Jesus, who he is and what he's done; to focus on God and his character - his faithfulness and truth; and to leave the work of the Spirit to the Spirit.
But the question itself also deserves an answer. If you're deliberately trying to be something you know you're not before others then that is concerning. It doesn't mean you're a full-out fraud but at the very least it indicates that others' estimation of you looms too large in your thinking. That's something to examine, as to why it matters so much. There are many possible reasons for it, which are a whole conversation in themselves, but, again, one of the best ways to handle it is to see yourself as united to Christ and accepted by God in him, the only one whose acceptance of us counts for anything.
The saying referred to is helpful up to a point, that point being the one we've just made. But there's more to say. We all have different temperaments and personalities. Some are very gregarious, others far more introverted, and still others have aspects of both in their makeup. So we ought to expect variations in our preferences and in what helps us. Blanket statements are good for giving a general picture but they fail at parsing particularities.
But, beyond that, the Bible itself points up the blessing of praying together with other people. Without minimising Jesus' instruction to 'go into your room and close the door' when we pray, it isn't everything the Bible has to say to us about prayer. Our Lord himself was overheard praying by his disciples. The apostle Paul could speak about the prayers of Epaphras for the church in Colossae. Instructions to prayer are often given in the plural (for a church as a body).
That the prayers of others become something we find we can 'inhabit' in a prayer meeting, finding encouragement and the enlarging of our own heart - even the expression of our own heart - is not so far removed from experiencing the same thing when reading the prayers in the Psalms. We ought to expect that our hearts will be knitted to those of our brothers and sisters at a soul-deep level when we pray together.
And finding that sometimes, perhaps often, our appreciation of God and our sense of access to him is heightened in those times is not something to make us ashamed of our personal struggles but, rather, should encourage us to persevere in seeking God for ourselves.
